How do I transfer blog to new .com domain?

I just signed up and paid for domain hosting with MyDomain.com. However, now I'm not sure how to transfer my blog (hosted at tumblr.com) to the new domain so that it stays and gets redirected there

Put simply, you can't.
Install WordPress › Blog Tool and Publishing Platform onto your new server, and use that to blog.

About transferring your old posts, is the blog too big to just copy and paste them over to the new system?

cut copy paste?
other then that you can't