The PlayStation 3 has a fully integrated online environment for you to play games, download new games, and do other activities. Here is everything you need to know to get started.
First, every base PS3 unit needs to "update" before it can go online. This will be confusing. First, make sure you have your PS3 plugged in or wirelessly connected to your in-home internet connection. I have MANY articles on this site about how to get a PS2 / PS3 unit connecting to your web system, either in a wired or wireless manner. Read through those articles to make sure you have those basics set up properly.
Now use the PS3 console interface to go online. It will tell you that you need to update. It IS actually connected at this point, but it doesn't have the updates needed to talk to the PS3 interface on Sony's end. Go into the update screens and it will download those files that it needs. There will be a long download period, and then a long update period. Now you can register for a PS3 account.
You create either a master or sub account. This has to do with access abilities. Start with a master account. Scroll to the bottom of the giant legalese page. When you enter your information - email, password etc. If you have a USB keyboard this is relatively easy. Note that it is REALLY annoying that your password "stars out" AS YOU TYPE IT. Talk about non-helpful!
There are twelve pages worth of avatars that you can choose from, to represent yourself.
Now you're set! You can use the online interface to sign in. You can now add friends, block people, look through your message box.
You CAN surf the web with this, on your high definition TV screen! However it can be *really* tedious without a keyboard, depending on what types of pages you go to. You will have to point-and-click to get to the first page at least. Once you do that, if you're just clicking links and reading, it's not too bad. If you're going to forums and typing in messages, one letter at a time with the point and click interface, it will probably drive you batty.
As far as a specific game going online, you'll need to read the review for that specific game to see what it can support.