I posted earlier stating that I saw Google Play Services listed under my apps, and it was up-to-date as far as I know. It does ask me if I want to install it but then takes me to the Google account sign-in (which I can't sign into.) Clearing the data on every Google app I saw and rebooting did not resolve the sign-in issue.
I'm going to try to re-flash the ROM, fix, and radio because I managed to sign in during the set-up. I'll see if that clears up all of my problems.
EDIT: WORKED! Few minutes in and not a single problem so far. Signed into Google and the Play Store works. I may be speaking too soon though.
Thanks for the help!
Side question: How can I remove some apps such as HTC Apps and HTC Backup? I click on them and it says the apps are not installed but they still show up in the list of apps.
You can hide them in the app drawer.