I wouldn't go so far as say you made a terrible mistake. I would just say that for a tic better sound, in addition to other features you don't have with the stock head unit, it would have been better to start with the head unit and then move to the speakers.
I went with a Pioneer, double-din touchscreen. It's the
AVH-280BT model and sells for a
couple hundred bucks, but there are many single-din options that would provide nice signal and power at a lower cost.
Yes, you can add an amp to increase the power going to your new speakers however, the head unit is the starting point of signal processing, so you're still not going to get as much out of those speakers as you would with a cleaner signal from a good aftermarket head unit to the amp and then to the speakers.
If I were you, since you've already purchased the speakers, I'd install them and then work toward getting a decent head unit next. You will notice a significant bump in sound quality, in addition to gaining a host of other cool features like Bluetooth connectivity, etc.
Search and lurk the
In Car Entertainment + Electronics (audio / video / alarm) section. A lot of good info to be had there (and is where this post should have been created to get the most relevant help).