To obtain Green Lantern's Ring there are a few options, these include

Defeating a Raid.

Finding Abin Surr's Crashed Ship (Roughly 15k blocks from spawn).

Randomly Obtaining a ring after roughly 2 hours of gameplay.

Getting a constructed one from a Current Green Lantern.

You also get a Green Lantern Battery from all of these options, you can also get one through a unpowered lantern battery and a Green Lantern charging it

To obtain Yellow Lantern's Ring there are a few options, these include

Obtain Bad Omen & the Darkness Effect at the same time whilst being low health.

Randomly Obtaining a ring after roughly 2 hours of gameplay.

Getting a constructed one from a Current Yellow Lantern.

You also get a Yellow Lantern Battery from all of these options, you can also get one through a unpowered lantern battery and a Yellow Lantern charging it

The Black Lantern is different from the rest of the Lanterns, When you first load into the world you have the chance to get a specific emotion cap, If for some reason all of these emotions are super low (sub 10k) you will begin to be called by Nekron to do his bidding, It's not known what Nekron could be looking for but we assume it's some sort of Soul, possibly Human.

When becoming Nekron's Vessel you gain the Black Lantern Scythe & Ring, This ring isn't charged like a Normal Lantern Ring instead you need to drain the life out of players and mobs, This ring also cannot construct anything but Black Lantern Rings.