I'd just like to point out that as much as people like to think otherwise, the vortexes can be used as dedicated dps bombs for shadow and elemental (and their neutral enemy types work nicely enough, too; especially with the suction effect). Their damage from them is only very slightly lower than blast bombs, and the charge time is about a half second longer, but it does gather the enemies up for you, ensuring that the maximum number at once can be hit, and keeps them together so you can clump up more and more for spamming of the bomb.

Thanks to the fact that it gathers and damages crowds easier and quicker than any blast bomb (especially with the blasts unleashing huge KBs requiring you to reposition yourself amongst the enemies) their overall dps output is actually a lot higher than it might sound on paper. Even with their KB effect, it's not very extreme and leaves them ready to funnel back towards you into the next vortex (where they get quickly sucked in again anyways) and the electron has the handy shock effect.
To finish it off, hazes combine nicely with this as well, since all the monsters are gathered together; your haze can be guaranteed to affect as many as possible. It's also very helpful if your hazes aren't 5* yet and don't have the full sized rad.

As well as being a great, safe and fun bomb to dps with solo, it's also very team-friendly, as it gathers up a crowd for your party members to unleash their favourite charges into as well.

Vortexes may be classed as support, but they can most certainly dps with the other bombs.

To get a look at what I mean, I made this demo video about a year ago, showing clips of vortexes being used in RJP.
(note: the video does include some other bomb footage that you may or may not be interested in. The dedicated vortex stuff is about halfway through. Although, since the OP is asking for bomb advice, it may help)
Also, if you're new to bombing, I also made this bombs demo video for bombers in the making, around the same time, which gives a more general look at bombs. Note, however, that anything regarding sun shards is now out-of-date, as the old RSS no longer exists.

Like Darkbrady says, both vortexes are excellent and must-haves for a dedicated bombers. Personally, I started out with Ash of Agni and the old Sun Shards.

Your priority should be a little something like this:
-Spine Cone and upgrades are all-around excellent for taking out healers. It's a great damage bomb for everything but slimes and contructs.
-You will be the most popular guy in your guild/group if you have shivermist. All the blitzers will want you on their FSCs!
-You will also be popular if you have a vortex. The pre-5* bombs are too short and tiny to be of much merit, but it's very worth it to get it up to that point. Not only is it great in a group, but they're decent solo bombs in their own rights. They combine well with almost anything.
-You will eventually want all the haze bombs. Voltaic Tempest is a great crowd controller and Venom Veiler can make a room full of gremlin healers a joke.
-If you're at all interested in the expansion pack, the Dark Retribution is a lot of fun and, despite complaints from some people, pretty powerful.

For a startup bomber, the choice is pretty much between Volcanic and Mad Bomber. Volcanic gives CTR and fire resistance, while Mad gives CTR+Damage at the expense of being very vulnerable to statuses. Can you deal with being shocked or on fire for about 15 seconds at a time? How much do you really need that extra bomb damage?

Also, get a Volcanic Plate Shield and never worry about shields again.

Got to agree with Fradow here, as I've been saying the bombing thread for a long time now, Shiver is inferior to Vortex in the vast majority of cases, and out-performed by other status bombs in many others. Especially with the recent Stun-fix making Stagger Storm an extremely good bomb, Shiver's utility is really dwindling. The only places I even recommend taking Shiver are FSC for locking down Vanaduke and Trojans when appropriate, and Compound 42 to better control the huge numbers of Toxoils since Shiver not only immobilizes them but also puts out their fire (Plus they're actually immune to just about every other status once ignited). Aside from maybe also using it in the Shadow Lair version of IMF there really isn't anywhere else you're ever going to want to use Shiver.

As for a generic bombing loadout although it's recieved ammendments in recent patches this is the go-to loadout;

Volcanic Demo / Mad Bomber Helm
Volcanic Demo / Mad Bomber Suit
2x Elite Boom Modules
Stratum-appropriate shield
Dark Briar Barrage / Dark Retribution
Graviton Vortex / Electron Vortex
Ash of Agni / Voltaic Tempest / Stagger Storm / Venom Veiler
Ash of Agni / Voltaic Tempest / Stagger Storm / Venom Veiler

This covers just about all your bases, although as is the case with any bombing loadout you can swap in Nitronome or Big Angry Bomb over any of them if you feel the need to. Shivermist is obviously good for FSC and if you're planning to run FSC a lot you'll want to make it, but it's dead weight everywhere else. Shard Bombs aren't worth prioritizing, make more important weapons first and then grab some shards if you want to try them out. Irontech Destroyer and Heavy Deconstructor are basically collector's items and pretty much useless.

In turret waves Shiver only really helps more than Nitro will if your team is un-coordinated. Otherwise you all go to different corners between waves and with Nitro chaining you can lock 2-3 of the turrets entirely until they die. Although yeah Shiver is undeniably useful in the Vanaduke fight (Although whether that's explicitly because of Shiver's utility or the fact pure-bombers really have no alternative to actually contribute there) but elsewhere -even in most of he citadel itself- other bombs are better and it's important to stress since a lot of newer players tend to end up idolizing Shiver and learn to rely on it, which really hampers on improvment and generally speaking is a rather literal definition of a crutch. Although flimsy-skill arguments aside the most important thing really is a lot of end-game vets find Shiver boring, and as bomber I find Shiver spamming mind-numbingly uninteresting compared to the myriad of more visceral, effective and extremely pretty explosions I could instead be making.

I mean I think the important thing to stress here is bombing as an art is all about specialization, you definitely want Shivermist, Venom Veiler and any other "niche" bomb just because exploiting those dynamics if what makes bombers special in the first place. But if you're asking me what order I recommend someone makes bombs, Shiver is going to always going to be put off as long as is possible because -especially for newer players- it's prone to generating bad habits and in many ways limits someone's perspective on all the wonderful quirks of array of bombs avaliable to them.

The graviton bomb is valuable for two reasons:
1. It can't shock like electron, so it's preferable against quicksilvers.
2. It's shadow damage, so it'll kill gremlins and slimes faster as long as you chain it.

However, I still think Dark Retribution is a better shadow damage bomb. It's also excellent against constructs.

Skip the focus module, as full mad bomber and level 10 bombs will leave you at Max CTR anyway. For the same reason, you won't need more than a single boom module. You could still have the focus module if you want full CTR even before reaching heat level 10 I suppose.

FSC:
Electron Vortex
Shivermist
Dark Briar Barrage
Voltaic Tempest/Stagger Storm

I like comboing EV and Shiver, as the shiver, as mentioned in my last post. I also like following that up with sun shards or crystal bomb but shard bombs really are just for the particularly interested.

IMF:
Graviton Vortex
Stagger Storm / Shivermist (In the shadow lair, Shiver can be used to put out oilers, but it also makes gremlin bombers spastic)
Dark Briar Barrage / Dark Retribution (I did the Shadow Lair last weekend and there were a lot more gremlins than I expected. I think the run would've fared better if I'd brought this.)
Some kind of gun, for the switches.

GV/Shivermist combo, or Stagger Storm + 3 x Damage Bomb

RJP, aka the best place to bomb ever:
Dark Retribution / Nitronome
Graviton Vortex
Ash of Agni
possibly Venom Veiler

GWW:
Dark Briar Barrage, for everything
Electron Vortex
Voltaic Tempest, for spookats
A sword or gun, for the boss. Bombs are suicide there.

Perhaps a better bomber than I will correct me on IMF and GWW.

Either you get a heart trinket, which can also be used for Lockdown, or you get an offensive trinket for your side-weapon, which can also be used if you change role sometime in the future.

DBB is preferable if you play in a team, not only in FSC, because DBB is team-friendly, Nitro is not.

In RJP, yes Graviton is useful. One useful bomb he left out is Voltaic Tempest. The shock dispatch minis like no other due to it's aoe property, all the while being good support.

And yes, the GWW loadout he gives is for SL mode : there is no kats in the normal mode.

Do AoA instead of Nitro. It's more party-friendly and ice cubes are super weak to fire. Like Fradow said, Venom Veiler is the more redundant option as, honestly, Ice Queen is even easier than the regular Royal Jelly.

For the combo I mentioned, I place and EV and then immediately a shiver. Provided you're fast enough, the shiver should detonate shortly before the vortex explodes, keeping the enemies all clumped together.

Regarding Lockdown, it's true that VT is a powerful tool there for captures, provided nobody on the enemy team are immune to it. I tried a few matches recently after I got my mercurial armour and I was surprised to find how much more effective it made shardbombing there. Suddenly I have that little edge I need to dodge hits while weaving through my own shards. Shard bomb are, however, very tricky to use and you will need a lot of patience and training with it.

About Stagger Storm, it's a generally good support bomb that can be used absolutely anywhere. You can replace Shivermist, VT or Venom Veiler with stagger anywhere you feel like slowing down the enemy would be more helpful than the other effects. Notably, Darkfire Vanaduke is immune to Shivermist and as such you might feel like Stagger Storm is the better option to bring there.

As for soloing with a sidearm, a good loadout imo is full mad bomber with a boom mod, slash mod and BTS. This gives your bombs max:max stats and vh sword damage, so if you bring along a fast sword (or even a slow sword with KB) it allows you to easily keep yourself safe in a quick situation without needing to charge up a bomb, since stopping to charge isn't always ideal.

It's also much better for bushes and grass~

1) I would absolutely recommend bringing a non-bomb for IMF. It doesn't have to be a sword though. In general, If you're cornered you could just shield-bump, or knock the enemy back with Valiance.

Back in the old days I used to bring a 2* calibur along with my 4* bombs for IMF. These days I have a few 5* guns and swords I bring as sidearms in some situations. Examples of why you might want other weapons than bombs with you:
-Multiple greavers can be tricky to bomb, but a little running in circles inside the radius of a DBB mostly gets them.
-Areas with a lot of switches, such as IMF, can be made a complete chaos if you're using bombs. As long as you have teammates to hit those switches, you could rely on using small-radius bombs such as vortex and haze bombs.
-Some distant switches might need guns. You can get around this using vials.
-Minerals take a while to break without bombs other than DR. However, minerals are dumb and nobody cares about them.

2) I've never heard of this. Do the fireballs usually unspawn on their own?

3) I don't know, really. I think ice queen's health simply doesn't scale as much as the knight's damage do at the lower depths of shadow lairs. For some reason, Ice Queen and the Red Roarmulus Twins seem to die faster in the Shadow Lairs than on regular boss runs.

4) Shard bombs are awful on Vana, use DBB. In the third wave, the shards won't even have the reach to reach through both layers of fireballs. You might also want Nitro for the mask, as it is the weakest to normal damage. In fact, if you had to choose between just Nitro and just DBB, I'd bring Nitro for a bombs-only FSC.

Have heard it myself but never tested it. Should try it out.

In fact, I need to spam FSC today; will grab Zeddles and test it today to confirm~

EDIT: Just tested it. Myth Busted. Doesn't even so much as reduce the rate at which they respawn~