Hey. The A1200 has a better chipset and a newer Rom version for the operating system (AmigaOS). Some old stuff doesn't work. You can go to Aminet, wich is the largest deposit of share/freeware stuff for the Amiga and get a downgrader, wich is a program that will make the A1200 "behave" like olders models.
Alternatively you can get WHDLoad wich is a program to install old games on harddisk. There a few stuff to read but should be fast and easy if you read the docs!
For old games there are some sites around that have thousands of old ones for download (they requested permission from original company/authors). If you use a PC for download you'll need to somwhow transfer the files after that. You can use a serial cable or a PC formated disk wich the Amiga can read (not the other way though, PCs can't read Amiga disks). There are various ways, and again do some search, read the docs and should be straightforward. After that, use WHDLoad to install the game on the Amiga's harddisk or simply save it to floppies (transdisk is the program I use).
Sorry if I say read the docs a few times, but it's just too much stuff to write, specially since it's the kind of question that pops up in here many times. Maybe someonw will have more patiente than me :-)
One obvious cool thing to do, would be to upgrade your Amiga and get it on the net directly.
Cheers