You want a good pair of parallel clamps or pipe clamps too, for when gluing stuff, which can hold something precisely for a long time with a lot of pressure, but most of the time, you want something fast.
I also recommend a few clutch-style bar clamps, since a lot of time, you're reaching over something.
If I got pipe clamps would I need to have a whole range of pipes just to handle standard work?
Pipe clamps are comparatively cheaper with the bonus that if you need longer clamps, you can go to the plumbing supply for a longer bit of pipe. They aren't as nice as parallel clamps for gluing up thin panels though.
For handscrews, I highly recommend Miro Moose. The Besseys are surprisingly subpar. Lie-Nielsen makes them as well for a reasonable price, but they galvanize the screws, and I don't like the action as much.
Any advice on what to look for or what brands you might recommend?
I bought a couple furniture clamps a few years ago and they are complete garbage.