I've got a gunman character, and it's definitely harder, especially at the beginning. but now I've got 4 points in Firearms and Expert now, so I'm much stronger than any of my followers. I'm still using the Quality Revolver I got from my bandit background though. What gun should I try to get next? Also, when, and with what weapons, should I go to the Black Mountain Mines? I'm level 17 right now, and have 4 followers.

You can use your fate point to critical success in pickpocket on tarants gunsmith/trader. He's holding pretty good revolver. But the best guns you get through schematics.

Do you have any points in Gunsmithy? If you have enough, you could build a Hand Cannon, Looking Glass Rifle or Elephant Gun, which are potent weapons. If you don't have enough, a Repeater Rifle works well, but uses up a lot of ammo. You might also see what schematics you can buy from gun smiths. Depends what sort of gun you'd like to use.
I suggest holding onto that Quality Revolver as a backup, as its speed can come in handy if you have a slow main gun.
As for when to go to Black Mountain Mines, you're ready. Pick a gun and pack plenty of ammunition.

What kind of armor is good for a gunman? Does wearing heavier armor (non magical) have any detrimental effects?

Heavier armor requires more strength, or you'll become encumbered and lose action points. Noise Penalty can also be a problem if you're a more stealthy character.

Machined Plate is the best armor for a tech character not using stealth, but is bugged. It's supposed to boost strength while worn, but doesn't. Even without the strength bonus it's good armor, but is heavy.
The Unofficial Patch fixes that and lots of other bugs, and adds some new content. If you'd like to use it, it can be found here, in the Arcanum slot.

Though it does begin by being hard, using the non fight methods of quest completion you can quickly progress to decent guns and then its just a matter of upping perception, intelligence and dexterity. If you are having trouble at the start though you can get off to a flying start with a little pickpocketing skill (lvl 1 is fine) and gun-smithing
In Shrouded Hills
1. Stand in the junk shop
2. Fast forward time to late night
3. Shop keeper will go to bed after time is fast forwarded, follow him into his room
4. Once he is asleep, pickpocket the key
5. Fast forward till morning, shopkeeper goes into the front of the shop
6. You are free to use the key on the chest and steal all his junk to make guns & grenades
7. If you need something not there just keep stealing and fast forwarding time until you get what you need

What's the best tank companion to get for an evil gunslinger? I just need someone who can sustain indefinite amount of damage. He/she doesn't even have to be able to deal any since I'm doing quite a lot of damage myself.
There's one orc I know but he doesn't like evil characters...

Check out the Ancient Temple northeast of Tarant in the Morbihan Plains for a follower. I won't give the name unless you want me to, but it should be obvious enough.

To chime in on the getting-good-guns subject, if you don't want to spend points on gunsmithing, you can procure the Looking Glass Rifle in Ashbury from a peaceful certain someone, and there is an Elephant Gun that's held by some bad dudes you fight later on in the game, in the quest you get from the cold-blooded dudes.
The latter, while later (comparatively) in the game, took my character from being pretty awesome to being pretty much unstoppable. The former is awesome as well, especially since you can snipe with it from crazy far away.

Definitely put seven points into gunsmithing; not only can you make the Elephant Gun, but once you find the schematic in the shops you can make the Blade Launcher (fine dagger + elephant gun), which is even more overkill. I didn't even have enough strength to properly wield it, and I was still stomping everything in my path. Chews through bullets like you wouldn't believe, though; six bullets per shot. If you insist on not spending points on gunsmithing, you can also get a Mechanized Gun in Caladon about halfway through the game from a man called Maxim. You have to do a quest for him and then bring him a Clarington Rifle and a Loading Mechanism, which is good if you have abundant supplies of ammunition - 6 bullets per shot, speed 20. If you find your ammo supply is a problem, you could probably stick with the Elephant Gun instead of upgrading to the Blade Launcher.
After that, it's a toss-up between Explosives and Electricity. Any gunslinger should have at least one point in Explosives, because it allows you to make your own bullets, and the Molotov Cocktail is easy to produce and makes a great crowd control weapon. Some of the higher level grenades are pretty good as well, though I wonder whether it's worth putting seven points into explosives; Dynamite takes too long to detonate to be useful in combat, and planting it near an enemy is likely to cause him to attack you, so I don't see it having much use in combat. For demolition, a couple of shots from your Blade Launcher should smash any door or chest in your path.
Electricity might be worth it because it gives you some of the best tech gear in the game. Electro-armor, Flow Disruptor Shields, and Charged Rings in particular. My suggestion is that if you intend to play a charismatic character and build a large party as I did, go with electricity over explosives. It will help you equip them better. If you plan to play more of a lone gunslinger-type, the explosives should come in handy. Either way, though; put two points into Electricity for the Charged Ring. Build two of those and equip them to get a four-point boost to Dexterity, which saves you two points for something else. The Tesla Gun is also fairly good for gunslingers going up against anything resistant to normal damage, which is a lot, especially towards the end of the game. Having both the Tesla Gun and Blade Launcher allows you to switch off between batteries and bullets if you ever run out of one, but I found it easy enough to just stockpile bullets and give the Tesla Gun to Sebastian. I used a mod that makes all ammunition weightless, though, so weight would probably become a big concern for those not using that mod. A lone gunslinger might wish to put some points into Mechanical as well; you can make your own traps and minions.
As for normal skills, definitely max out Firearms (of course). Pick Locks is reasonably useful to save you the trouble of bashing down doors, but I don't think it's really worth the points. I suppose it'd be useful if you intend to steal from shopkeepers, though. Persuasion is good for a charismatic character; Expert Persuasion increases your max followers by one and there's at least one (rather annoying) dungeon that can be entirely bypassed by a persuasive character. Probably worth putting at least enough points into Dodge to get Expert. Otherwise, put your points into Perception and Intelligence, as well as Charisma if you intend to have a large party. Maybe a few points to Strength to get the best out of your elephant gun/blade launcher if you have points to spare, but you can still use them just fine without a high enough strength stat. I think you take a penalty to firing speed, but with guns that powerful, does it really matter?
Definitely recruit Jayna for healing (Virgil and other magical healers will be useless once you raise your technical aptitude). She may be useless in combat, but it saves you a few points that would otherwise have gone into Heal and Herbology. Her Therapeutics skills are pretty useless, but keep her well-stocked with healing items and she'll do a fairly good job of keeping you and your party alive. Sogg Mead Mug makes a wonderful tank, and he can be recruited fairly early on. I found Magnus to be fairly effective in combat as well, especially once I gave him Sebastian's Charged Axe (which you should definitely acquire, even if you don't bother to recruit Sebastian himself. It's one of, if not the best axe in the game). The Dog is also pretty effective in melee combat, but some argue that he's too powerful, so you might not want to recruit him if you're looking for challenge.
If you find the early game too difficult, try turning the difficulty to easy. it's still tough, but it's more doable. Strangely, I found the game actually got easier as I played, especially as I started to upgrade my party and their equipment, so you might be able to put the difficulty back up later on.

Despite the possible spoiler, how and/or where do I get ahold of the blackpowder and bullet schematic?

I found the bullet schematic from one of the Tarant's merchants at random.

Yeah, the gunsmithy in tarant usually has one.
I don't know if this counts as a spoiler, but I'm about to say where you can find the ingredients for bullets. Charcoal can be found in the trash or bought for 3 gold from the general store or the gadget store, and saltpeter is available in the same places. Nets you 20 bullets, which is about half the price of buying them, if you don't have high haggle, charisma, etc.

If you don't mind melee at low levels, 2x charged rings + balanced sword + melee 1 can get you through the first third or half of the game until you get enough stuff to make a decent gunner. Heck, I was ripping things apart with a balanced sword for almost the entire game.