re: Starting Woodworking, what should I get?

For cabinets like that you will want to build them out of plywood and then trim with wood. A biscuit cutter makes that job much faster and easier.

For the vertical pieces you will need to make dados. You can do that easier on a table saw but a router makes pretty quick work of them as well. Just have to take a little more time measuring and layout prior.

The Kreg jig is a must.
A good drill/driver should also be high on your list.

lastly, be sure you lay out everything prior so that you have good access to studs when installing.