Electric dryer squeaks and rattles, but still dries and sometimes stops working, but if I let it sit for a while, will turn itself back on. Is this something I can fix myself, or should I just look at getting yet another used dryer?

If your dryer is less than 10 years old it is probably worth repairing. Your problems sound relatively minor, but call a qualified repairman.

If the squeaks and rattles are not objectionable, then you should at least consider.

A thourough cleaning of anywhere lint can hide in the dryer. This may cause it to turn off.

All rotating parts need to be lubed.