Depends a lot on what you want to test. There's not really any generic tool that you can just pop on a chip and see if it's working.
Say if you want to test an opamp, you'll have to put it into a specific test circuit. It's a lot more complicated than you think.
Also, with some ICs you just can't test them, it's virtually impossible. Most of the time you'll just have to power it up, probe some pins with a scope, test rails, and then narrow it down. Then you'll just have to guess what needs replacing.
If you just want to test components, you'll need a good multimeter and LCR meter. If you want to do anything more, really you'll need a good digital scope, and a really good understanding of the system you're working with.