why did my record player slow down, speed up, then go back to normal?

Didn’t spend enough on your cables and contact cleaner; )
is it belt drive? Did the belt ride down on a crowned pulley possibly?

Self check?

Occam's Razor: the simplest answer is always space aliens.

If the table has a speed control dial it most likely needs to be cleaned with contact cleaner.  These are sometimes located underneath the table or some have controls on both the top and bottom that each need to be cleaned to keep consistent speed.

I actually went to play a record today, and I had to press the (electronic) buttons multiple times before it did its function. There was no speed swell, though.

I am thinking that the buttons are corroded. 
Also, @stereo5 I did the talcum powder thing today. Thanks again for the tip!