Musket Ball/Endless Musket Pouch
+costs nothing to use
-no distinguishing traits
You might find yourself using the Endless Musket Pouch for practical reasons; it's an infinite ammo source, so it's the best ammunition to use for exploration if you want to save the more valuable stuff for important fights. Note that several guns gain the High Velocity piercing effect when using the Endless Musket Pouch, including the Uzi which is available before you normally have access to those bullets.
Silver Bullet
+no farming required
+absolutely destroys Werewolves
-no distinguishing traits outside of that
These earlygame bullets have the extremely funny effect of dealing triple damage to Werewolves, which might be more relevant if Werewolves were a common enemy and not restricted to a single moon phase.
Tungsten Bullet
+no farming required
-no distinguishing traits
Basically, Musket Balls with 2 more base damage. Unlike Silver Bullets, they retain the High Velocity effect when used with the Uzi, giving it marginally higher DPS than the Endless Musket Pouch while retaining its piercing properties.
Meteor Shot
+pierces or ricochets once
-mostly nonrenewable
A piercing bullet you can use early on, though you are generally encouraged to use arrows rather than bullets in situations that require crowd control. Still, if you're a gunslinger you probably found this extremely useful against the Wall of Flesh, and it'll be pretty useful for quite some time afterwards.
Assuming that the average meteorite yields 300 pieces of ore, a fully-mined meteorite can be converted into 7000 Meteor Shot, which is enough to feed an Unreal Uzi (the least ammo-efficient gun) for more than 15 minutes. So use them carefully, but don't be too stingy.
Exploding Bullet
+high knockback
+damages multiple enemies
+no farming required
+good velocity
The absolute best anti-Destroyer bullet, with some utility outside of that when facing dense crowds too large for Meteor Shot to handle. They don't require an upgraded Anvil to craft, can be made in bulk, and drastically increase the viability of several early hardmode guns.
Party Bullet
+decent knockback
+no farming required
+good velocity
-no distinguishing traits
-messy
The best generic bullet you can make in early Hardmode without farming materials, though that isn’t much of a selling point considering how easy it is to farm for better ammo such as Crystal Bullets.
Golden Bullet
+increased profit when used on bosses/minibosses
+no farming required
+good velocity
-no distinguishing traits
The best way to use them is for repeated Eye of Cthulhu and Queen Bee kills, and they won't even work for that in Expert Mode since bosses drop treasure bags instead. They're not for use against things that actually threaten you, except maybe for dealing the final blow to Pirate Captains.
(mythril or orichalcum anvil)
Crystal Bullet
+highest potential DPS
+large number of hits
The best bossing bullet if you don't think accuracy is going to be an issue. They are more likely to deal extra damage when fired against targets that are moving towards you. Crystal Bullets have a great synergy with Ichor and easily shred enemies with low defenses.
Cursed Bullet
+high damage
+good velocity
-no distinguishing traits
Dealing a fixed 24 DPS to any target stuck by a Cursed Bullet doesn't seem too significant at first, but when there are loads of enemies on the screen all burning at once, that can add up to some significant total damage. Of course the passive burning damage does nothing to control their movements, so be careful not to get overwhelmed.
Ichor Bullet
+high damage
+inflicts a powerful debuff
+good velocity
Generally superior to the Cursed Bullet if you're packing any sort of competent offense. The Ichor debuff greatly enhances DPS, so it is generally the strongest bullet if you don't have another way to inflict Ichor. But you usually do have another way, so it's hard to recommend Ichor Bullets when you can deal much more damage by using a whip, dart, or Golden Shower to inflict the debuff and using Crystal Bullets instead.
(one mechanical boss)
High Velocity Bullet
+incredibly high velocity
+pierces twice
+no farming required
The synergy between the Endless Musket Pouch and Uzi/Venus Magnum/Sniper Rifle is good enough that you generally won't find it necessary to craft an actual High Velocity Bullet if you own one of those guns. If you don't own them, you can go ahead and craft these bullets for cheap, which mainly benefits the Megashark (the most accurate automatic gun in this tier). Chain Gun enjoys them, too.
(all mechanical bosses)
Chlorophyte Bullet
+perfectly accurate
+uses a highly contested resource
Arguably the best bullet in the game. There are many guns that depend on Chlorophyte Bullets in order to be viable, and every gun is greatly improved when using them. Highly recommended for boss fights, or any situation really.
(plantera)
Venom Bullet
+high damage
+no farming required
-no distinguishing traits
Basically a straight upgrade to Cursed Bullets. However, they will usually deal less damage than Crystal Bullets, despite their higher base damage.
Nano Bullet
+high damage
+smart bounce
+inflicts situationally useful debuff
+no farming required
The positive effects of smart bounce aren't fully realized during aerial battles, but when fighting underground or near the surface, you'll find these very useful for their accuracy boost. While not quite as reliable as Chlorophyte Bullets, a major point in their favor is that you can just buy them. They also have the same base damage as Venom Bullets.
(final boss)
Luminite Bullet
+near infinite piercing
+highest base damage by far
-uses a highly contested resource
-comes too late to be of any use
The last bullet available. It destroys crowds of enemies and provides comparable DPS to Crystal Bullets, but won't work as well as Chlorophyte bullets against things that fly and move around a lot. You're not hurting for DPS given that you unlock Vortex stealth at the same time, so these bullets might be overrated, honestly.