Thanks for all the input, I'm starting to go through all the suggestions. I noticed today that when I probe ground pins on the ICs, there is some serious noise going on, in some cases +/- 150 mV. The VCCs all very flat and seem fine though.
The clock circuit etched into the board turned out to be incorrect so I made a separate circuit and wired it to the bottom. The clock output looks fine and I have used it in an old Atari 2600 with success. The solder mess on the top of the PCB is where the old clock circuit was and doesn't seem to have any shorts.
I also added a copper board connected to the grounds of the ICs. Just to see what would happen, I put a random capacitor I had laying around between the ground plate and +5 V. I'm still getting a ton of noise with these modifications. Perhaps I should try to power it with an external supply?
I'm sure the power supply circuit is horribly inadequate, I copied the simplest 7805 based circuit I could find (schematic here:
http://i.stack.imgur.com/yeKoT.gif ). This whole project was rushed and limited by the free version of Eagle's restrictions, and of course my lack of knowledge of PCB design and Eagle cad. Once I got the autorouter to spit out a 100% completed layout, I went with it! I could post the schematic and board layout, but they are very
very ugly. Rather than designing a new chip icon for each new library, I just copied a library with the same package type and re-purposed all the pins, so the ICs look a bit off. Also there are no wire connections between anything, only points I named and connected. There are no component values on the schematic either.
Attached is a picture of the bottom of the board, which probably looks hilarious to some of you guys
It is a prototype after all
Anyway, thanks a lot for everything, hopefully we can get this thing working!