I have a Shimano 8 speed 11-34 cassette (one of those "mega range" ones). If I change my cassette to a different one, say an 11-32, will I need to change or otherwise modify my chain? I just got the chain replaced and now it's slipping on the old cassette. I'm not the biggest fan of the mega range, so I wouldn't mind changing it out to something different.
As the smallest cog is the same and the largest cog is smaller, the gear range is a subset of the old one. The old chain length should give you no problems at all. If you were going the other way a longer chain might be needed.
Theoretically, you could shorten your chain to accommodate the smaller (largest gear) cassette. However, I never install a new cassette unless I install a new chain as well. Cassettes and chains tend to "wear" to one another and one can sometimes encounter difficulty if you don't.
That "could" be the issue with your new chain. If the old chain was run past it's proper service life and damaged the cassette, no new chain is going to work well with it. Depending how many miles/hours you put on the new chain, if you put a new cassette on your rig and leave the "new" chain, it could be fine.
However, there are many other issues that could cause slippage. A stiff link in the new chain if it was twisted/damaged during installation would be a prime example.
As there is only 2 tooth difference on the top end and the same number of teeth on the bottom end, you should not need a new chain or adjustment. There is some room in the derailleur to adjust the length (between 11 teeth and 34 teeth). Your new range of 11-32 is within the 11-34 so there should be no problems.
The one thing you would have to make sure is that the chain and cassette are compatible, correct width, size, shape. So if you are using Shimano, your new set should also be Shimano of the same groupset if you want to make sure it will work.
Generally Yes you have to change your chain. The only time you might get away without changing chain is if its JUST been replaced. That is, under a hundred km of riding or in the last ~week.
Running an elongated chain on a new cassette dramatically increases wear on the cassette.
It really depends on how long/far since "it was jut replaced"