Before you go looking for games you need to make a few decisions:
1) Are you looking for free games or are you happy to pay for them?
2) Do you want a game that can be downloaded to your iPhone or are you happy to play online games?
3) Are you hoping for a full console game or are you happy with the usual type of games that are available for cell phones?
There are free games available for iPhone – but most of them are played online. In other words, you do not download them to your iphone.
Playing games online means having free games, plus if you know where to look you can have a good range of games to choose from and there are some games where you can compete against other players.
Downloading games to your iPhone means you can play them anytime – even if you do not have access to the internet. But they do take up memory!
Whilst there are some great free games that are available, it can be quite tedious to find them on the internet as there are so many spammy sites that have to be waded through before you can find the good sites with games. Some sites do specialize in finding them for us and I recommend you keep some of these bookmarked and just keep checking back for their latest updates.
If you want to download games to your iPhone then you have the options of either buying them through Apple (simple and easy to find) or through a membership site (not always easy to find these sites but much cheaper).
If you are wanting the "full console" type games then you really will have to pay – in the short term – for these type of games. They are still some months away from being available and it is possible that then the membership sites will also have them available which will mean considered savings. So I recommend you check them out before you spend a small fortune at iTunes!