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

The reverb on your Fender amp isn't working.

Well the flux capacitor has to charge up fella..

Cap charge, my guess. Lose power connection.

Every time or just once in a while or JUST one time?

Regards

Too little info, so all would be a guess. On start up, while playing, how often, type of drive....

First, is it idler drive, belt drive or direct drive? If idler, perhaps the idler wheel needs resurfacing, if belt the belt could be slipping. You can put the belt in a bag with talcum powder and shake it up and reuse the belt. If it still slips, get a new belt from the manufacturer.  If it’s direct drive, the speed controller circuitry could be defective.

Brief power outage or surge, bad capacitor, belt slippage, bad bearing, severe stylus drag, finger on the platter, platter rubbing on plinth, Schoenberg, bad karma for having been a troll.