Well my favourite series aren't the most esoteric ones so I'll add in some other stuff after that you might not have heard of that might be interesting.
Favs:
The Legend of Zelda: Action adventure series in which you adventure around the land and make your way through dungeons fighting enemies, solving puzzles, getting cool items, and eventually making your way to the boss. I recommend Ocarina of Time, Majora's Mask, Wind Waker or Twilight Princess. Most of the rest are also really good but those are the ones that really stand out to me. So far Skyward Sword hasn't really inspired me much.
Forza Motorsport: Simulation racing series exclusive to Xbox/360. Great attention to detail with a wide range of cars and lots to do with them (like upgrading, tuning, painting etc.), cars feel great to drive, great online community, and lots of assistance options means pretty much anyone can play the game at a level they will enjoy. The latest game (4) is the one you should go for. It's the best of all the ones on 360. The original on the old Xbox is also still really good. Obviously the game has worse graphics, less features, and a more basic driving model, but it has a lot of cool tracks that never made it in to any of the 360 games.
The Elder Scrolls: Pretty hard to not know about this series since Skyrim came out, but I'll talk about it anyway. Huge expansive first person RPGs with hundreds upon hundreds of hours of content. Basically if you are bored of your own life and want to throw it away and start a new life in a fantasy world, you can't do much better than Elder Scrolls. I've played 3 (Morrowind), 4 (Oblivion), and 5 (Skyrim). All are great games but I wouldn't recommend Morrowind to someone who is new to the series. Very hard to get in to and more convoluted than the other games. Also does very little to help the player. For example there's no compass or map markers for quests, you have to work out where stuff is yourself.
Other stuff:
Darksiders: It can't really be called a series yet, since only one game has come out so far but there's another coming out this summer. The game is basically Zelda but done in a post Biblical apocalypse setting. Also the art direction for the game was handled by a famous comic book artist (Joe Madureira) so it has a very cool unique over-the-top visual style to it. The combat in the game is kind of a light version of something you would see in God Of War or Devil May Cry, but is very satisfying. The first game isn't the most expansive game. There's little to do apart from the main storyline, but the second seems as if it's going to be far more expansive, adding in many RPG elements and making the world far larger.
SSX: Ridiculous over-the-top snowboarding series. Really cool colourful visuals with insane tricks to pull off with a great soundtrack playing in the background. So far only 1 game in the series has come out in this console generation, which was on the Wii. But that was total crap. A 360/PS3 game (simply called "SSX") is set to come out this spring, and so far it looks awesome. I recommend trying out SSX Tricky, 3, and On Tour.
Devil May Cry: Action game kind of like God Of War, but has actually been around for longer. Gameplay is very similar in that it focuses mainly on deep, dramatic looking combat which can get extremely tough if you choose it to. Has a kind of demonic, Gothic style to it, but with a Japanese flavour. I haven't had loads of experience with the series but I hear all the games are really good apart from 2, which was abysmal. There's a reboot coming out later this year made by a different studio than before (Ninja Theory). Lots of people are complaining about this being a Westernisation of the series, but so far I think it looks pretty awesome.
