The most confusing part of the game for me so far is exactly when players can take 'player actions.' Is there an easy way to explain or list all the times when taking a player action is possible? I feel like I'm doing something wrong during the challenge phase in this regard.

Download the FAQ here : http://www.fantasyflightgames.com/edge_minisite_sec.asp?eidm=10&esem=4

Near the end there are graphics showing the processing of each phase. So use it while playing at the beginning.

Shorthand - Player actions before challenges are declared. Player actions after attackers are declared (but before stealth is declared). Players actions after blockers are declared (but before challenge resolution). Player actions after the Challenge.

I would definitely look at the FAQ though, it breaks it down in more detail and their are some interesting bits about first player you will want to learn because of how they affect what I wrote above.