Just copy the entire website to the new computer.
If you just want to move the project from one computer to another, just copy it. If you restore the database on the new computer just as it was on the original computer, there should be no reason to manually
change the connectionstring. All of that is stored in App_Data and will move with the website.
If you are performing an Orchard upgrade, say from 1.5.1 to 1.6, this is a completely different scenario than your question. In that case, check the release notes on how to properly upgrade an Orchard Website.
Based on the following thread, the error may be caused by a database permissions problem, but I am not familiar with the error personally.
I recommend using your favorite search engine to search the internet for generic ADO errors like the one you received, which is what I did in this case. If you are getting database errors ( orchard cannot connect to the database ), that would explain why
the pages aren't showing up. Double and triple check your connectionstring, your credentials, and that the database is working correctly. Try to connect to the database with that connectionstring and credentials outside of Orchard CMS to be sure. Sounds like
a connection problem.