The basic gunslinger armours start out at 3*, with the Sash and Stetson. I generally find that the universal gun damage bonus is more useful than ASI bonuses of the same strength - ASI can be obtained from the Swiftstrike Buckler, several armours, trinkets, and UVs, whereas the universal gun damage bonus can only be gotten from a couple of armours, and trinkets. (Other than Shadowsun, the only other non-trinket equippable that gives a medium damage bonus to guns is the cheap 0* Cyclops Cap.) The Seerus masks are great if you can get them, giving resistance to two of the more dangerous statuses as well as CTR Med and ASI Low.

You're going to want the specialised damage guns. Some guns are better to start out with than others.

• Drivers are pretty much the bread and butter of a gunner's damage. At 5* they have a badass charge that has the capability of OHKOing turrets when fully heated, even in parties. This is all down to their special ability, the ricochet; if you hit the right spot on a monster, then it's possible to get the ricochets to happen inside rather than bouncing off, resulting in MASSIVE DAMAGE. They're also useful for clearing out mobs, too, since closely-clustered monsters will have the ricochets bounce off each other. The most favoured ones are Nova and Umbra (pure damage) and Storm (shock having good mob-clearing abilities with its AoE damage). Hail is less commonly used, since ricochets can break its status, but works reasonably well as an utility gun. There's also Magma, but I am less of a fan of this one because the DoT inflicted by fire doesn't really make up for the loss of damage most of the time.
• The Pulsar lines (Polaris and Supernova) are infamous. On the one hand they're excellent at herding enemies suppressing turrets, since you can shut them down just by using a steady stream of expanded shots. However, they also have slow bullets and result in a great deal of knockback and screenflash, which can be irritating for allies to deal with if they're engaging enemies in close combat. Neither should be used for the sake of damage. Having elemental damage and shock status, the Polaris tends to be more popular than Supernova as an utility gun against turrets. Please be careful when using these.
• Autoguns are quite hard to get used to. On paper they look like the best things ever, especially the Blitz Needle, which has one of (if not THE) highest damage outputs in the game. However, you need to unload an entire clip (6 bullets for standard attack or 15 for the charge) at once, they come with the added drawback of locking your knight in place whilst firing, which can potentially be fatal, and it's possible for your enemy to move out of the cone of fire. Best used for killing large, slow enemies, such as Trojans and Vanaduke, and especially where you can reliably predict the spawn points (since this saves time and health charging up. Can be used against Beasts and Undead. Best not used against Devilites and Greavers, since they will generally dodge out of your cone of damage/dash towards you before you can finish charging properly, unless you prepare beforehand. Blitz is pure damage. Volcanic trades out pure piercing damage for fire and lower normal damage, with a bunch of knockback on its charge (more useful for RJP and IMF runs against the bosses). Plague trades out some piercing damage for the poison status, which is of debatable use and takes a lot of LD tokens, but looks way more badass.
• Antigua line guns are not really a gunner's weapon, oddly enough. They make good sidearms for sword-users or bomb-users, but the damage bonus vs. families is rendered redundant if you're running with a maximum damage bonus for guns. They fire fast with a large clip, and are mobile with their regular attacks, but do low damage per hit. Their charges leave you stuck in place, like Blitz, but with less damage and a bird that pushes stuff around. Great in combos with vortices or on stationary turrets, but poor otherwise.
• Catalysers are odd, and pretty much the opposite of the Pulsar lines. Instead of normal shots that expand and explode, you can 'tag' an enemy with charge shots, before detonating them all. In practice this sounds awesome, but the charge shots expire after a minute, and the bullets are slow, making it easy to miss with charge shots. (Pulsars are the same speed but don't have the same issue, since their shots expand to ridiculous size, making it easy to hit with them.) There's also the issue that using them in a PUG will mean some sword-user will kill your walking bomb before you can blow him up. Neutraliser is normal damage, making it good for use on zombies which make great walking bombs; Biohazard is shadow damage with poison status, which means it's extremely useful in RJP and its Shadow Lair variant, as well as on jellies of all kinds. Both of them also look badass. Fun, if not especially practical as a first choice.
• The Magnus lines aren't really damage-efficient guns. Callahan is a good utility gun for hitting dodgy enemies such as Devilites, but only in small numbers, since it locks you in place, making it do less well against groups. It inflicts stun, has piercing damage and knocks down 'soft' enemies such as Devilites, Wolvers and Gremlins, flinching them in small numbers. The charge is useful if you want to commit seppuku in PvE. Iron Slug is badass-looking, if pretty useless. It deals normal damage and doesn't get a status or damage bonus over its brother, while it still keeps the drawbacks. The normal damage means it can cancel turrets, but given that the Pulsars can do the same job a lot better, most people don't bother with the Slug. The Callahan is somewhat useful for utility, but not excessively so due to the small clip and restricted mobility.
• Valiance is the gun of your dreams. And by that, I mean it's pretty much what you'd expect a gun to be like. You get normal damage, with no fancy bells and whistles like knockdown, piercing, ricochets, explosions or whatever. Just three fast bullets that fire instantaneously, and do a bit of pushback on every hit. The charge is a bigger and more damaging version of the standard bullets. It's good for knocking stuff around or as a decent general-purpose sidearm, and is reasonably mobile.

tl;dr: I would recommend you make a start with drivers, and go from there. I would also recommend getting a Blitz at some point for pure damage purposes, as well as going after the damage-bonus armour. Learn how to dodge so you can use a Swiftstrike Buckler (without dying).

The drawback of the Polaris is that it is not party-friendly.
Your bullets are going to knock things everywhere, which may include knocking enemies into your allies; nothing causes offence quite like a Lumber to the face, after all. The flash can also cause issues on computers with less powerful graphics cards, as well as obscuring parts of the screen. The Polaris doesn't have the high-damage ricochet ability of the driver series, either, but that isn't really an issue. If you play solo as a gunslinger, though, the effects of Polaris won't be a problem for you.
In parties, as long as you remember to use Polaris as it's meant to be used - as an utility gun, rather than just wantonly using it for damage purposes - then you will get on well with most people in the Clockworks.
Storn Driver and Umbra Driver will work well together as a duo.
Polaris and Valiance won't really get you great DPS, though.
Since you only have 2 weapon slots open, you're going to want all the damage bonus you can get, so your guns should be 2 different specialised damage types for best effect.

Polaris damage (unexpanded/expanded/charge, shock tick)
Weak: 159/191/368, 28
Neutral: 92/130/306, 22
Resistant: 28/32/59, 9
Damage values taken in t3 training hall against dummies, equivalent to depth 25.
Weapon fully heated with no damage bonus UVs or equipment.

Edit: ...huh. Oddly enough, Nova Driver has the same basic damage values as the Polaris, with the expanded bullets being equivalent to Nova's regular shots. That's assuming that you don't get any ricochets in with the driver, but even so.

" what's the drawback of the Polaris"
- visually disruptive
- huge random direction knockback, which will makes you miss your own shots and may dirsupt your team too (a lot) => that's the deal breaker
- not a very good charge

"Sure, the bullets are slow, but is it the big problem in PVE? "
I wouldn't call that a big problem of Polaris (Biohazard/Neutralizer have that problem), due to the expanding bullet which greatly reduce the need for correct aim and planification

"how many damage makes the Polaris on short distance"
Wiki : http://wiki.spiralknights.com/Polaris expand the damage table and have a look at the numbers you are interested in.

"But for that i can use Umbra Driver or Valiance."
Valiance is not recommended because it's normal. You'll be better served by 2 non-normal guns : that way you'll have bonus against 4 over 6 families.

Blarg. Typed all that and forgot to post, Bopp and Aureate already answered.

Edited, sorry Aureate. Batabii used to have that avatar for a very long time, so I tend to associate it with him.