+1 on the chains/cassette.
Old cassettes tolerate newer chains at some level. Old chains typically do NOT tolerate new cassettes.
In the old days guys would have two chains and swap them out weekly or monthly. But with the pins and stuff that's not really practical.
To check your chain, put it in the small ring and a cog you use a lot (or one of the smaller ones) and, while sitting, push very hard while going slow. The chain will jump if it wants to. Don't stand else you'll sit really hard on the stem or top tube, not fun. Check all the gears while in the small ring, and if things seem fine, you'll be fine.
cdr