Dukeplatypus gives good answers, including these, but I'll throw in mine too.

Are swords still the "go-to" kinda weapons? When I was playing, swords were the only thing you could really use as a main weapon, if you wanted guns you had to have 4 different guns and you were still less effective. Bombs were for the crazy guys who just wanted to mess around. Can you now use bombs or guns as "main" weapons?

You could always use guns and bombs as main weapons, although bombs were a little tricky for novices. When using swords or guns (or a mix), you never needed more than 3 slots, to hold piercing, elemental, and shadow weapons. You also don't need more than 3 slots when using bombs. My main knight always uses 3 slots, and my secondary knights always use 2. No problem.

Is the Wolver (I think that's what it's called :P) line still the "best" armor?

For sworders it's the best. For gunners, the Gunslinger line is best. For bombers the Demo line is the best. As Dukeplatypus mentioned, the Chaos set has recently been buffed, to the point where it's the best at everything --- except defense. Consult the Armor section of my detailed sword guide.

If I don't want to have 4 weapons (just keep the two slots) is it best to have one sword/gun or 2 swords/2 guns? If I remember right, Neutral damage was the simplest and worked "ok" against all monsters, but should I have, say, a gun for certain types of monster and a sword for other types? Like a gun that's strong against Gunpuppies and a sword for close up monsters like Jellies? Or are some monsters just more dangerous and so I should get weapons to kill them fastest?
Looking at the wiki and damage types, if I get a Elemental gun for Gunpuppies (because I hate 'em) and a Shadow Sword for Slimes (because they come in hordes, if I remember right), then I should use my gun on Constructs, Undead, and Fiends, and use my Sword on Slimes, Gremlins, and Beasts, right? That way I'm strong against 2/3's of the game and neutral on the other third. Is this viable?

Yes. Consult the Mixing Weapon Types section of my sword guide. A piercing sword and elemental gun is a little stronger than a shadow sword and elemental gun, because fiends are really nasty (and piercing swords are pretty good against gremlins anyway). But shadow sword with elemental gun is also good.

Has much really changed?

Missions were a big change for most people, but you can still do the same old Clockworks, and the game is still delightfully fun. Welcome back.

To address your question more directly: Last May the damage numbers for guns and bombs were suddenly increased by roughly 15%. So they are more competitive with swords now, than they used to be. But, as we've said, playing with them has always been practical. For example, an alchemer in the hands of a master gunslinger is pretty potent. And Blitz Needle is pretty potent, even in the hands of a novice.

Bombs and guns were always strong enough. But they might need strategy to work properly

Swords are just easier to get used to. Run towards and enemy and slash it. But that doesn't mean they are better.