The easiest way is to use the Project Archiver, under the File menu. This will move your project as well as the media files in your project to a new location without losing the links between them.
BTW, you're not also moving from an old version of the program to a newer, are you? If so, I strongly discourage your opening an older version project in a newer version of the program (eg, opening a project created in version 7 with Premiere Elements 8). This almost always leads to buggy behavior.
I strongly recommend you finish your project in the same version of the program you started it in or, at the very least, finish editing the current segment and use Share/Personal Computer/AVI to export it as finished video file and import that into your new version's new project.
Better safe than sorry!
In you old computer, where are the Project file, the Assets and the Scratch Disks located?
In your new computer, where will they be located?
PrE uses absolute Paths in the links, so those Paths must match 100%, or then one needs to help PrE locate the files in their new location. This is really easy, and is semi-automatic. PrE will ask, "Where is file _____.avi?" and you just navigate to that new location. It will then survey that folder, and link to all other Assets in that folder. If you have multiple folders, you will need to help PrE with each folder, but it will re-link to all Assets in that folder too.
Now, if you are planning on installing PrE on any other HDD, but C:\, I recommend against that. PrE and PrPro really want to be installed onto C:\ and run as an administrator. The Projects, Assets and Scratch Disks can be on any other HDD.
I also second Steve's suggestion to finish all Projects in the version of the program, where they were started.