I have experience with this. Changed a freewheel on a bike, but not the chain, which was stretched. 11 months (!) later, the freewheel was garbage.
Measure your chain with an old style American ruler (with 'inches'). Twenty links should be exactly 10 'inches'. [0] If it's 1/8" 'inch' longer, replace the chain. If it's 3/16 'inch' longer, replace the rear cogs and the chain both. Consider replacing the front chain rings, too, especially if the chain is longer or they're made of aluminium.
A stretched chain ruins gears. Ruined gears will ruin a new chain, too. Proper maintenance fixes both problems. New quality chains cost about US$10-15.
[0] Wikipedia says one 'inch' is 2.54 centimeters, if you believe that.
My most recent experience with new-chain/old-cluster was that the chain skipped a tooth three times per pedal revolution! Replaced gears, problem solved.
To go buy the new gears, I first had to temporarily put back the old chain, to make the bike ridable. :)
Adjusting the derailleur can help; in particular the "B tension" adjustment to get the jockey wheel to track the gear cluster more closely for more wrap. However, if it's that bad, it won't help.
I have decent experience adjusting derailleurs. Even taking apart the non-adjustable ones and drilling holes in new places to change spring tensions.
> 8-20 of them
You're not going to get 8 teeth of contact on an 11 tooth cog, nor 20 teeth of contact on a 26 tooth ring.
I still wouldn't consider a cassette something you should change as a matter of regular maintenance. Certainly not as often as a chain. I've had cassettes last through multiple wheelsets.
I've never heard of chains slipping teeth, most notably because it would need to slip all teeth in contact at once (8-20 of them).
If you are having your chain slip off, you need to adjust your deraileur.