The List

This is a list of things to do before you quit Eve.
This list is not exhaustive, so please, if you have any suggestions, leave them in comments.

Hunt some belt rats in:

-High
-Low
-Null 

Go mining

Orbit Faction Warfare Plex

Low-sec PVP - Simply the best PVP there is.

Visit the Eve Gate
Duel Rixx Javix
Run a sleeper site.
See a black hole.
See Chribba's Veldnaught
See Mittanigrad
Duel Jack Dancer in a Breacher.
Ninja mining in low sec
Find a Mordus legion ship
Manufacture something
Gas mining
Set up a POS
Create a trade mini-hub
Go mining in a wormhole
Tag ratting in low sec
Exploration and find something valuable
Get a navy faction battleship
... Fly a battleship

Ships to fly (in PvP):
Garmur
Wyrm
Omen
Gila
Rifter
Breacher
Slasher
Condor
Kestrel
Merlin
Incursus
Tristan
Atron
Punisher New Punisher
Executioner
Tormentor
Slicer
Firetail
Hookbill
Harbinger