Today's ride confirmed that the drivetrain problems encountered in my previous two rides were, indeed, caused by a worn out chain. The problem of the chain trying to constantly shift to the small chainring went away with the new chain installed.
This new chain does not shift very well on the rear cogs. I've tried adjusting the derailleur, but it doesn't help. Since it shifts poorly in both directions, adjusting really means I can get it to shift well one way, then not at all the other. Right now it's about as good as I can get it and I am hoping it'll get better if the chain wears a little bit. It did seem to get better towards the end of today's 35 mile ride. Since I paid $50 for it, I'm stuck with it for about 3000 miles.
So, with today's 35 miles and yesterday's 65, I had a 100 mile weekend.