Why would car still just be clicking and not turning over at all after I just replaced the battery?

you might have shorted out something-started, but the same thing happened to me, and it was the stupidest thing in the world and wouldn't hurt to check, do u have an alarm system? my car has a trip sensor so when the battery is changed or shorted to try to hotwire, alarm is automatically applied, and wont start with the alarm on

The clicking sound is the starting relay and is typical of a battery with low voltage. You didn't state the reason the old battery was replaced. If it was because of age, the above alarm issue may be the cause. If the old battery was only a couple of years old and not holding a charge, there could be a slow drain from short.

If it worked when first installed and within a few starts it is failing, then I would suspect something in the charging circuit. A good alternator will put out about 14 volts at the battery terminals with the engine running.