I own and have extensively played all of the Civ games. IMHO, the order of ranking is:
Civ IV > Civ V > Civ II > Civ I > Civ III
Civ IV with all the expansions is still my favorite Civ game to date, it's just a complete, enthralling, awesome thing to play. That said....
Civ V is the version I play today. It has the best interface and graphics, although I greatly miss the charts and diplomacy screens from IV. I feel that after another expansion or two V will surpass IV as the best Civ game. God's & Kings made V much better, and I've now pretty much shelved IV for good.
Civ I was orgasmically awesome back in 1994, mine is on a single 1.44MB floppy, but by today's standards its very dated and only good for nostalgia.
Civ II was an improvement over Civ I in every way, it's even good today IMHO.
Civ III is in my opinion the worst game of the series. That's not to say it's bad or even terrible, it added some real nice things to the franchise, it just fell short on the fun factor and always seemed lacking to me. I eventually abandoned Civ III and went back to Civ II before Civ IV came out.
IMHO, don't bother with III, you won't be missing much.