You do not want to export and import photos.
You want to export and import catalogs.
One one computer, open LR, select the desired photos (which could be all photos) and then File->Export as Catalog. If you check the box next to Export Negative Files and then you can transfer the original image files as well as the catalog.
One the other computer, open LR, then File->Import from Another Catalog, and point to the location of the catalog you exported in the previous paragraph.
DJ's solution is a good one. There is another option that may work for you too. It is possible to save the metadata of files to the hard drive. If you save the metadata externally, all of the work you did to the image will come into LR with a new import. So if you have a folder of images in catalog A, and you save all of the metadata externally, you can then import that folder to catalog B, and all of the edits and keywords will get imported too.
Metadata can be saved manually by selecting an image and doing a CMND or CNTRL S. Or you can turn on 'save metadata externally' in the preferences (I'm not in front of LR at the moment, but I think it's found in the first tab). If you turn on the preference, the metadata will start to save for everything in your catalog. This could take awhile with a large catalog.
Theresa J wrote:
If you save the metadata externally, all of the work you did to the image will come into LR with a new import.
I have to disagree with the word "all". Collections are not transferred this way; virtual copies are not transferred this way; and a bunch of other items are not transferred this way.
Yes that is true. My husband often accuses me of exaggerating. Guilty as charged
The external saving of metadata is a good solution though if only develop settings, ratings and keywords are needed.