What tests should I run after a new build?

You could benchmark your pc using 3D Mark, amongst other things..

You can also test stability using "Orthos" which pushes you CPU/Memory to the limit.. Also there is CPU burn-in..

I run Prime95 burn-in. Prime95 is a tool that uses a distributed computing technique to find very large prime numbers. One of it's side uses is to put stress on a CPU in various ways. There's some examples included in the readme available on the download page on the link above.

this is the best (Free) CPU burn-in utility I know of

When doing a burn in. It tests your computers parts at max. If you have any bad part or parts the tests will tell you which ones. And it also lets you know how stable a computer is. Now most parts that are going to fail will fail in the first 24 to 48 HRs. Now there are other thing it does for you but I'll let some one else go in to it. Just know it is a very good thing to do on a new build.

Hope this helps