Are any of my files going to get messed up/deleted?
Not if the upgrade goes smoothly, if it doesn't due to a unknow glitch, then yes, your entire Macintosh HD partition may require erasing to fix the issue, which will also erase your files.
So you need to backup personal data off the machine before inflicting any sort of major change.
Will games/apps not work anymore?
Programs using PowerPC processor binaries old older Mac software will not run on 10.7 or 10.8 anymore, however will do so under 10.6 using the supplied Rosetta that gets installed automatically.
Some apps/games are not updated or tweaked for newer operating systems as it will cost a lot of money to do so.
Some apps will require a paid upgrade.
Will upgrading slow down my computer?
10.8 and 10.7 (and 10.5) are slower than 10.6 freshly installed on the same hardware, so yes, upgrading to 10.8 will slow your machine's performance down, less so if it's freshly installed (no other data on the hard drive.)
10.8 was designed for newer faster hardware with more features, it thus runs slower on older hardware, even slower on less powerful older hardware.
If your machine is older than 3 years and running 10.5, I say go to 10.6 and stay there.
If your running 10.6 on a machine less than two years old, then your a good candidate for 10.8.
If your running 10.7 already, then 10.8 isn't going to make too much of a difference.