I'm re-reading my organic chemistry book to prepare for a lab class, and I've taken a quantum chemistry class. I don't remember too much from quantum chemistry. In a general chemistry sense, is there a way to determine a number of bonds an atom can make just by looking at its position from the periodic table? For example, without quantum chemistry knowledge, how should I know that phosphorous can make 5 bonds? Is it just something to be memorized until quantum chemistry?

Have a look at the bottom of this page.

The predictions are not perfect (for example 6 valent oxygen doesn't happen) but they are a very good start.

http://www.chemprofessor.com/ptable.htm