I did some more tests (with my limited knowledge of hardware) and yes, 555 square waves are not perfect SO I tried using the really nice square wave coming out of the CAL port of my ancient $40 oscilloscope and while it's better, I still get those spikes.
I then tried to smooth out the slightly spikey looking square wave from the 555 by adding a large pot after a 100nF cap on the 555 output so I could adjust it and see what happens. It gives slightly rounded edges to the square wave until eventually I could make it look like a curvy triangle-ish wave. I'm assuming it just a simple lowpass filter of some sort like I use in synths all the time.
The slightly filtered square wave output still produces a spikey (but better) triangle wave out of the integrator circuit. I was using a roughly a 2.5kHz signal.
QUESTION: What's with the spikes that show up at the inverting input of the opamp?
Here's what a square wave ends up looking like when I probe the inverting input of the opamp. It turns into a flat line with little spikes along it's length.

This is what it looks like when I probe both the Input - and the Output pins and set my oscilloscope to A & B

If I set my oscilloscope to A - B of course it looks like what I want it to because it cancels out the spikes in Channel A from Channel B.

Like I said, it gets much better if I use the nice clean square wave from the CAL port of my oscilloscope.
