...Generally I would recommend an elemental alchemer, Argent Peacemaker or Polaris. All of them do at least neutral damage against every current turret type, and effective versus two out of three (Howlitzers and Gun Puppies).

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Elemental guns:

• Elemental alchemers are capable of doing lots of damage to clustered mobs or single enemies due to their ricochet ability, but have a small clip and can set off explosive blocks and switches by accident. The shadow version functions exactly the same way; the status versions all get less pure damage, but get a status to make up for it. Storm is most favoured out of the status versions.
• Argent Peacemaker is mobile and has a large clip as well as fast bullets, but each bullet does low damage so you're going to need to keep firing a lot to get a high DPS, which will result in sore trigger fingers.
• Polaris is the troll gun - it can easily suppress turrets with its shocking and attack-cancelling abilities, as well as heavy knockback, but the last (in combination with its obnoxious screenflash) can result in much rage. Its bullets expand midway along their path for extra damage, and are very slow. They also explode much like grenades.

Other guns:

• The Autogun lines are all high-damage weapons, but have the unfortunate added trait of locking you in place whilst firing. Blitz is generally favoured for dealing with trojans, Deadnaughts and Vanaduke, as well as fiends, beasts and pretty much anything nonresistant to piercing. It's best used on monsters you're extremely familiar with; unfortunately two out of three turrets are resistant to pierce, so it's not worth getting it for those. Volcanic Pepperbox (normal damage and fire status) is sometimes used instead, although the locked whilst firing mechanism puts many people off it for general use, and it has lower base neutral damage than the Blitz. Plague is almost exactly like Blitz, except with less pure damage and a poison status, and its recipes are only purchasable with Krogmo Coins since they are non-tradable. (It also looks sexier, but I'm biased.)
• Valiance is a good all-round sidearm. Its bullets do decent normal damage with some knockback, and are fairly fast. Despite appearances, you cannot stab people with the knife on the end.
• Callahan is generally for dodgy monsters, such as devilites and gremlins. It does pierce damage, has fast bullets but a slow reload time, inflicts stun and knocks down enemies that it hits. Generally, this gun is used by people who are either trying to be hipstr or like playing with pure guns/more obscure weapons. It isn't a bad weapon, but its damage output isn't all that great and its knockdown ability pales when you're faced with several enemies. You also get locked in place whilst firing, but this isn't as big an issue as with the Autogun lines since you aren't forced to unload an entire clip at once.
• Iron Slug is like Callahan, except with normal damage and no status. Theoretically you could use this to suppress turrets, but in practise it isn't a great gun and is outdone by pretty much every other one. It is apparently getting a buff at some point in the near future, but apparently Spiral Knights has the same concept of 'near future' as Steam does.
• Supernova is the normal-damage cousin of Polaris. It shares the obnoxious flash and pushback as Polaris, but lacks any status. I'm pretty sure that the damage on neutral enemies is about the same for each gun, without taking shock into account. Not quite as popular as its elemental-damage cousin, but still a decent turret-suppressing gun.
• Biohazard and Neutraliser are fairly uncommon guns. They both have extremely slow bullets. Their charges orbit an enemy without doing damage, and can be stacked on one enemy; they only do damage when they explode, set off by a normal shot. They are much like bomb-guns, and are fun to play with, but not generally used since they're not all that damage-efficient. Attempting to use the charge in LD is ridiculously hard. The former gets shadow damage and poison status, and it is not infrequently used in RJP runs as a result, or on clusters of jellies. The latter does normal damage and no status, so can be used effectively on groups of slow-moving zombies and retrodes.

This is pretty much every gun I could think of at the time.

@Dark: ...you forgot trollaris
how could you forget trollaris D8

That's pretty hard, given some people seem to think it it is the only gun. (That and Blitz Needle.)

@OP: Oh yeah, you'd be best off getting a toothpick as well to round off that set (Flourish or Snarble Barb, depending on personal preference. They operate almost identically, other than the charge and crafting cost - the former is a sped-up, higher-damage version of the standard attack combo, and the latter is a bunch of spikes that fly and cover a larger area. The status variants are not as good for general use.) It'll help you greatly dealing with the monsters you mentioned.

I prefer the storm driver or sentenza when i usually gun for my friends on Vana and Jelly King runs or anywhere else i rarely use my Blitz Unless it is absolutly needed because you cant move when you fire and that makes you a target for wolvers and duskers and any fast monsters.