Installing Windows 7 on an XP machine isn't so much an upgrade as it is complete clean-slate installation. You'll need to re-install all of your programs and make sure that your supporting programs (Flash, Quicktime, etc.) are also installed and up to date.
Are you using the same version of Premiere Elements you used on XP? Only version 9 will absolutely run on Windows 7?
Also, did you install the 64-bit version of Windows 7? That too can cause problems, since many of the necessary drivers for your hardware may be 32-bit and may not work properly. (Though I assume you ran Microsoft's Windows 7 Update Advisor before you did your re-install.)
Are you able to open Premiere Elements? Start a new project?
There are a lot of variables here, so it may not be an easy fix.
An error comes up saying 'this type of file is not supported, or the required codec is not installed'.
This error usually indicates that a previous CODEC is not currently installed in the new OS. CODEC's are installed at the system-level, so just having your media files, etc., on the HDD's will not help here. This ARTICLE will give you more background, and tips.
What CODEC is required? That can be found by running the AV files through a utility, like G-SPOT. You need to find that CODEC, download it, and install it.
Once, Windows OS's included many common CODEC's, with the installation. With Vista, things changed, and especially for the Home editions. One had to go to the MS Web site, and download and install the Vista CODEC pack. Do not know exactly what is/is not included in the Win7 installation, but there should be an MS CODEC pack for Win7, where those were missing. WARNING: I do not recommend any of the "free" CODEC "packs," available, other than the ones from MS. Those often cause real problems, and I avoid them completely. At the very best, they fill up one's system with outdated, never-used CODEC's, that have to load, and usually will overwrite good, commercial CODEC's with hacked, or reverse-engineered versions. At the worst, they install all sorts of "stuff," and to get rid of some of that, one needs to completely wipe the system HDD, reinstall the OS, after a complete format, and then reinstall all software.
PS - let us know what G-Spot says the CODEC's are, and someone can point you to the one(s), that you need.