Even if you are connected to the internet you first must buy the Xbox live, which can be purchased as cards that can range from 3 months for $19.99 to 12 months for $49.99 (best deal).
Secondly, you do need a memory card to play xbox live. This is so the system information can be saved to the memory card so that it remembers your son's gamertag (identification) when he plays online. To play online, you need a wireless router and an ethernet cable. Most importantly, the wireless router, because the ethernet cable plugs into the back of the router, not the computer. If you do not have an ethernet cable that is long enough to run from the computer room to where the xbox is located, then you may need to get an Xbox 360 Wireless Networking Adapter which is $99.99, and because of the price, I only recommend this for hardcore gamers online, so if your son is young, then I do not recommend getting this. So in short you need:
Xbox love membership card
An Xbox 360 memory card
Ethernet Cable/Wireless Networking Adapter
Wireless Intenet Router
Also, since this is for your son, be aware that there are people (especially in shooting games) that use a lot of cursing/profanity. However, once you connect to Xbox Live, it will offer you an update that will allow you to set parental controls for your son which include- limiting his downloads from the Xbox Live Marketplace, who he talks to/is friends with, and the actual content he can play (like a program blocker for a TV.)
Good Luck...