I purchased apple care for my new computer, but had to return it and they sent me a new replacement...apple care is registered for a device by the serial number and now I have a new serial number, how do I get apple care for the replacement computer

I don't know if you need that. You should already have it. However, if the old serial number is on it, then they should send a new one after you make the change. If you have more questions about the transfer, it would be best to contact Apple directly.

Contacting Apple Customer and Support Service

1. Apple Store Customer Service at 1-800-676-2775 or visit online Help for more information.
2. Primary support help at Contact Support.
3. To contact product and tech support:  Contacting Apple for support and service including international calling numbers.
4. Starting an online chat with Apple Support.