I haven't played STO in a while, because of some issues with internet being really slow. Now that that's fixed, I wanted to try again.
During loading, my Windows Firewall pops up and blocked something called GameClient.exe by Cryptic. Can I allow this through my firewall, or is it something hazardous??
You should allow it. There should be two executable files of interest, one is the launcher and the other the game client. Ideally, you should launch the launcher first as it will check for patches and give you some notices and provide some other options. When you click "Engage", it runs the GameClient.exe file to actually play the game. You can skip the launcher and go straight to the game client if you so desire. (weird things may happen if you skip a patch though...)