They're obtained in parts of the main story; one in the early Sanctuary missions and one is unlocked :

when you get back to Sactuary after it takes off. If I remember correctly.

The Relic/Class Mod/Grenade Mod/Shield slots are only locked during the tutorial, they're unlocked extremely early. You never have to do anything but complete story missions to unlock the slots.

Weapons that REGENERATE ammo

Note, the following two weapons require a purchase of the Ultimate Vault Hunter Upgrade pack

Avenger

Counterbution revengence. The reckoning. – High overall stats. Behaves like a bouncing betty grenade when reloaded. Regenerates SMG ammo when wielded.

Butcher

Fresh Meat! – Very low pellet count. Greatly increased fire rate. Achieves maximum accuracy much more quickly than other Hyperion shotguns. Magazine randomly refills a random amount of ammo as it's fired, giving it a wildly variable magazine size in use. Fire rate highly increases under sustained fire.

Grenade Mods that REGENERATE ammo

The following 4 mods will require purchase of the DLC, Tiny Tina's Assault on Dragonkeep

Lightning Bolt

What's a jigawatt? – Always Shock. Regenerates grenade ammo over time. Shoots straight forward with a lightning graphic and explodes on impact. Character may shout "Lightning Bolt!" and a custom throwing sound will be heard after use while playing in Tiny Tina's Assault on Dragon Keep.

Chain Lightning

Don't pay it back, pay it forward. – Always Shock. Regenerates grenade ammo over time. Shoots a bolt of lightning straight forward that explodes on impact and arcs to nearby targets. Character may shout "Lightning Bolt!" after use while playing in Tiny Tina's Assault on Dragon Keep.

Fireball

Porkchop Sandwiches! – Always incendiary. Regenerates grenade ammo over time. Shoots a slow moving fireball straight forward that explodes on impact. When used, has a custom sound effect when thrown and the character may shout "fireball!" while in the DLC Tiny Tina's Assault on Dragon Keep.

Fire Storm

What manner of man are you that can summon up fire without flint or tinder? – Always incendiary. Regenerates grenade ammo over time. Shoots a fireball straight forward that explodes on impact. Four additional fireballs spawn above the spot of the original explosion and fall to the ground. Character may shout "Fireball!" upon use while playing in Tiny Tina's Assault on Dragon Keep.

Magic Missile

No wand required. Just point and shoot. – Always slag. Grenades slowly regenerate over time. Two (Blue rarity) or Four (Purple rarity) child grenades home in on targets and explode instantly. Character may shout "Magic Missile!" after use while playing in Tiny Tina's Assault on Dragon Keep.

Weapons that DON'T consume any ammo

Infinity:

It's closer than you think! (no it isn't) – Does not consume ammunition. Shots fired form a fixed lemniscate (∞) pattern, ignoring normal recoil behavior. Increased accuracy when aiming down the sights.

As an extra note, you'll need to have at least 1 bullet of pistol ammo, or the Infinity will not fire.

Some Vladof launchers:

Vladof launchers can have the special ability "reduced ammo consumption," which means that every third shot fired is not deducted from either the magazine or the player's ammo stock. This is coupled with lower base damage, though the overall destructive capacity of the weapon remains roughly the same as an equivalent launcher from most other manufacturers.