If you picked up a new Xbox 360 Slim, the first thing you're going to want to do is transfer all your old saved game data from your old Xbox 360 to the new internal hard drive in the Slim 360.
To do this you'll need a transfer cable. You can pick one up online at Gamestop, or from the Microsoft Store for about $15.
Doing so will allow you to transfer all your old saved games from the Marketplace or XBox Arcade, along with all your saved game data, music, and profiles. This process will work with any hard drive size (20GB, 60GB, 120GB, or even the 250GB if you had an XBox 360 Elite). Just use the standard Xbox 360 Hard Drive Data Migration Transfer Kit.
1. Plug the cable into the hard drive, then plug the USB plug into the back of the new Xbox 360 slim.
2. You'll see a prompt telling you: "Transfer cable detected. Do you want to transfer content now?"
3. Select "Transfer To Console".
4. You'll be promted to select what to transfer. If you just want to transfer everything at once, just select the "Start" button in the lower right of the prompt.
5. Just wait for it to finish and you'll be good to go.
Depending on how much data you have, it may take several minutes to complete, so be patient.
After you've transfered the content, you may need to transfer your content licenses. To do so, just follow the steps from Microsoft here:
Transfer Xbox 360 Content Licenses.