The best F2P MMORPG out there is
Dungeons and Dragons Online. By far. Yeah, yeah, you have to pay for most of the later dungeons, but playing through 1 character to lvl 8 or so on each server is enough to get you 2-3 dungeons that when played through on those characters get you enough to buy a few more. Only buying when stuff is on sale gives you even more of an advantage.
It's fun because almost everything is soloable at a high level of skill, but grouping is still fun and easy.
The downsides: no real PvP, endgame caps out reeeal fast. But I mean, it's completely Free to Play, so it kinda balances out.
A few MMORPGs to avoid:
Runes of Magic - shit WoW knockoff
Atlantica Online - fun premise at first, strategic grid combat in an MMO, but it's ridiculously grindtastic.
World of Warcraft - fun only if you have a group of friends going in, if not, don't bother and don't even bother trying to get to high level play in PvP as it's imbalanced as fuck anyway. I still bear no shame in saying it is STILL the Best MMORPG out there right now, it's fanbase and rigid structure benefiting only the truly hardcore are the only faults.
Lord of the Rings Online - Their version of PvP is even more ridiculously imbalanced, feels like an inferior WoW otherwise. Tries to focus on story but there really isn't any true depth and they're hamstrung by the established lore.
Shaiya - Everquest knockoff, PvP imbalanced as hell, classes extremely simple which is good for noobs, but not for decent level play.
For MMO competitive games, eh,
League of Legends is fun too. It's like DotA if you ever played that...