Well there are MMOs(Massive Multiplayer Online)games which are online games that are giant worlds where tons of people play together. The most popular is multiplayer battle arena called League of Legends. There is also the classic fantasy role-playing game World of Warcraft. A new MMORPG that is popular is Elder Scrolls Online, which is an online fantasy role-playing game. You could also play single player games on the computer like Elder Scrolls V: Skyrim, or Portal. There is an online game store called Steam, which is very popular and publishes all the new games and indie games. There are also websites like Andkon and Armor Games where you can play free flash games online
If those games don't work out, wii is really cheap right now, and I started with blackops on it. Now I have a ps3 and I'm A major gaming girl