I recently replaced a worn Shimano 9-speed chain and cassette with identical replacements. On my last ride, the chain would skip under moderate to heavy load when I was pedaling over railroad tracks, deeper bumps, etc. I think it was sliding quickly over the cogs, rather than skipping sideways. If it was jumping gears sideways, it was happening incredibly fast. Skipping occurred while I was in the smaller cogs. It didn't seem to be a problem on the larger ones.

What's going on? I didn't have this trouble with the old chain and cassette.

1. When installing the new cassette, I noticed I had to tighten the trim adjustment 2 full turns. That seemed odd to me since the replacement parts are identical, but the indexing and shifting now seems perfectly smooth.
2. My rear derailleur seems to return sluggishly around the hanger pivot, but cable tension seems appropriate, and the cable doesn't seem to be catching in the housing anywhere.

Any ideas?

Did you size the new chain correctly, that is, is the length correct?

I'd second making sure your chain is the correct length. A good indicator would be if the problem occurs more in your small chain ring than your big one. You said there seems to be some sluggishness in your rear derailleur? Your rear derailleur should have a really snappy spring too it. Is it Gunked up with grease and dirt? If the tension seems correct it sounds like the spring is pushing, but if its not snappy it sounds like something is gunked up. As to having to adjust your trim when replacing your cassette... from the sounds of things you have some maintenance issues. If you let a chain run too long when its in bad condition it will wear your chain rings as well as your cassettes. Eventually, your chain ring will start getting used to what we call a "stretched chain". Once this happens, its not uncommon to have to replace the chain rings, chain, and cassettes. If your rear derailleur is gunked up, it could be caused by over lubricating/greasing. There is very little point in trying to grease a derailleur as all the bearings are internally sealed. You would only attract dirt by doing so. Once you lubricate your chain, make sure you wipe all the lubricant off your chain (you only need to worry about lubricant inside the chain, lubricant on the outside does nothing, gets all over your bike, and attracts dirt).

The chain is the correct length. It's the same number of links as the old chain, which worked well.

The chain and cassette are brand new. I always replace the cassette when replacing the chain. The chainrings are worn, but not egregiously so.

I was boneheaded the last time I took my cassette off and didn't get the final cog on right and it caused that kind of problem. Not suggesting you could be as absent-minded as I was but if you're not sure ...