Best Unreal Tournament?

Personally it's Unreal Tournament: Game of the Year Edition for me. I did play more Quake 2 at the time though, you could play some Quake Live if you have the itch.

Both Quake & UT are great arena-based shooters. The later versions of UT offers more variety than Quake, as they also have vehicle combat (I think 2003 introduced vehicles, but 2004 absolutely had them) , potential for larger maps and a bit more variety in weapons design. That's not me saying that UT is better than Quake though. Quake is more fast-paced (UT is fast paced, Quake more so), and the level design focuses more on agility.
I've not played UT3, so I can't comment on it, but out of the UT games, I would recommend UT 2004 (ignore 2003, as 2004 has all the content of 2003 and more). Out of the Quake games, I actually preferred the first one, but Quake 3 has more maps and slightly tighter controls, so I think most people would recommend that one (and as others have pointed out, Quake Live is free),

I would say UT99 or Q2, since these are the only 2 I played more than for a while.

I played almost all of them and I can say that I love UT more than Quake because of the variety of game modes. UT2004's Assault mode is especially good with "automatically learning bots" and you'll get challenged according to your skill. After beating Godlike bots, you're ready for getting smacked on MP. (:

Assault, Mutant, Last Man Standing, Bombing Run, Double Domination, Onslaught... UT2004 can keep you forever. It has a team system and truly feel-alive bots that understands your commands. Best SP with bots I've ever played, hands down.

UT99 is similar but different. You should play it too. (:

I know I'm in the minority here, but my favorite was actually Unreal Championship 2 on Xbox. Loved the melee combat xP

They are all worth playing both really great series, they are both arena shooters but would say they play a bit differently. Are you wanting to play them for mostly single or multiplayer I've played a bit of Quake's & it's great but haven't had a chance to play much of unreal's yet. My Favourite Quake game currently is Quake II & favourite Unreal would have to be UT 2003. Quake is part of history too being the first 3d fps to my knowledge it's worth play just for that. Quake Live is free to play on steam too if you wanta try it out but holy shit it's fast paced

UT 2004 is the best one

I have played UT2003, Quake 3 and some earlier Quake games.
I gotta say that they all are great! I remember playing Quake 3 and UT2003 against bots all day. I had a blast with those games.
Recently I installed Quake Live (Which is basically an updated version of Quake 3) and I got a boner over how good it was. The old school shooter feel is something you don't get in Counter Strike or Call of Duty.
You already made a great choice by wanting to try either of the series out, so you can't do much wrong.

UT and Quake are both amazing games, played them when I was a little kid (3 years old, hehe). I still think Unreal Tournament GOTY is the best, I just love it. Tho if you want vehicles and bigger maps, go for UT2004.

UT99 for just arena shooter
UT2004 if you want vehicles along with space battles
Unreal Tournament 4 Alpha if you want something shinny

UT99!

UT2k4 is good too though.

edit: also haven't seen anyone mention it, but Unreal also has a multiplayer mode even if it's not branded with the "tournament" title (not that I'd recommend it over UT99 or UT2k4), just wanted to show it some love ;-;