Over a year ago I started on a fix-up of my 34 year old Rex Criterium and, with a lot of help from this forum and eBay, I took it out for it's first ride today (pictures tomorrow, I promise).

I thought I would go to the local rails to trails bike path near Gainesville Fl and ride a mile or so in each direction so I could set-up the shifters, brakes ,seatpost, stem height etc but it was such a beautiful day and the bike felt so very good right from the start that I rode the path front to back and then the return trip for a total of 32 miles.
Only issues I had (besides under inflated tires...too nice of a day to stop) was the chain jumps under pressure on two of the rear sprokets, the first and the third. The rear cluster is over 30 years old and has had a lot of miles on it so I guess it is some sort of wear but I can't see it...the chain is brand new.
What is this I read about cluster teeth being "twisted" to improve shifting..also the cluster seems to wobble a bit when I look down while coasting, is this a concern? Should I change the whole thing or just the offending sprockets?

BTW, if you are ever in North Florida this is a wonderful bike path...goes from Hawthorne to Gainesville.

Mike

I think I know the wobbling you're referring to. I'll bet it only "wobbles" when the freewheel is spinning free, that is, the chain is not driving it. I'll bet, when you put it on the rack and turn the cranks, the wobble goes away. Don't worry about that.
IMHO, replace your freewheel with all the gears. The new chain is telling you the gears are worn and the new chain is not seating correctly. If you don't replace the freewheel (with all the gears) you'll be wearing out chains prematurely not to mention putting up with that slipping in those two gears.

Correct on the wobbling,apparent only when not being driven. I had a feeling I was going to need a new freewheel and gears. Are they available piece by piece? I have a granny gear I would miss even though it is seldom used..I'll bet I'm gonna need some special tools. The hubs are old Campy so what do you call that thread pitch? Italian?

Mike

Pics as promised

A steel rear sprocket so worn the chain slips is something I have never seen. It's hard for me to imagine that. I've worn out several aluminum chain rings (teeth were just little spikey things) but never a single rear cog. I've worn out chains, too, if you accept a limited amount of stretch as the pass/fail criteria.

It could be that the freewheel is just shot. I've seen that more than once, with symptoms as described by the OP.

Twisted teeth!? Sounds like you've got a Uniglide cassette on that bike. I've had problems with those in the past doing the exact same thing your describing. Swapping it out for a Hyperglide should take care of it. https://www.sheldonbrown.com/k7.html