What version of Premiere Elements are you using on each computer and on what computer operating system is involved?
Have you given any thought to archiving the project from computer 1 (File Menu/Project Archiver/Archive or Copy option), and then transferring the resulting "Trimmed..." or "Copied..." Folder to a USB Flash drive for use in Premiere Elements on the second computer? No worries out Media Outline displays and Missing Media messages.
Please let us know if you need clarification on anything that I have written.
Welcome to the forum.
ATR has given you some viable options.
The PrE Project file (PREL), uses absolute Paths to link to the media in the Project. If any part of that absolute Path is changed, or missing, then PrE cannot follow those links. When you moved things to the second computer, it is very likely that the absolute Paths were broken, and most likely as drive letters changed.
One can duplicate the folder scheme in the second computer, if it has a drive letter available, that matches that on the first computer. As an example, if the media was on the D:\ on the first computer, but now are on the C:\ of the second computer, Moving those media files, in their exact folder structure to D:\, would likely solve the issue. If this is not a possibility, then I recommend letting PrE do the relinking. It will do this, for all media assets in each folder, once the first one has been located. This will need to be repeated for each media asset folder, but otherwise is automatic for ALL used media assets in THAT folder. This article on "Where is File ___ ?" goes into more detail: http://forums.adobe.com/message/4598705#4598705 Also see this article on Media Offline: http://forums.adobe.com/message/3561281#3561281
I often migrate Projects between computers, and for that task, I use external FW-800 (IEEE-1394b) HDD's. I first set the drive letter of each of those external HDD's to be the exact same, in the OS of every computer, that will work on that Project. Then, when I Import my media, usually Copies of the files, in each folder, into the Project, the absolute Path to that media is the same, including the drive letter. This article goes into more detail on using external HDD's: http://forums.adobe.com/message/4542396#4542396 This setup would only be useful to you, at the very beginning of a Project, and not once media assets have been Imported into the Project from a previous absolute Path. Also, this article on Folder Setup might be useful for the next Project: http://forums.adobe.com/message/4028231#4028231
I think that the easiest method to get your Project easily from one computer, to another, would be ATR's recommendation to go back to the Project on the first computer, and use the Project Archiver, to Copy that Project and all media assets, to an external HDD (remember to set its drive letter in the OS of both computers, before you start). Then, the links via the absolute Paths will be the same, and you should not get any Media Offline, or the "Where is file ____ ?"
Good luck, and please report your progress,