find out my apple id and password

Yes.

How to move data to your new Mac using Mountain Lion and earlier - Apple Support

For the question in yur subject line:

Is there a way to find my Apple ID Name if I can't remember it?

Yes. Visit My Apple ID and click Find your Apple ID. See Finding your Apple ID if you'd like more information.

How do I change or recover a forgotten Apple ID Password?

If you've forgotten your Apple ID Password or want to change it, go to My Apple ID and follow the instructions. See Changing your Apple ID password if you'd like more information.

One of the most trouble-free ways to migrate files to a different Mac or a different Hard Drive is to install a fresh Mac OS X on the "new" system, then bring your files over by using your Time Machine Backup. Otherwise Setup Assistant or Migration Assistant can be helpful. This very nice article written by the late James Pond shows you the options:

OSX Tips

Setting-up a new Mac from an old one, its Backups, or a PC