I broke the master link in my chain a while ago and my LSB told me it was because I wasn't cleaning/lubricating my chain often enough. The chain had experienced more wear than it should have as a result. I asked about replacing the chain and was told that you typically don't want to replace individual parts of the drive-train, that mixing and matching new chain and old chain-rings is a bad idea. Is there any truth to this? Why would a new chain on an old (we're talking less than 4 months) be a bad thing?
So, the shop guy may or may not have told you the truth, depending on how much your cassette is worn. This is not an easy thing to do if he didn't measure the chain with a gauge, so it is possible that is obvously worn, or he has a very good eye, or he actually wants to sell you the whole kit. Besides, the need to change everything together is surely NOT true, since the expected is to change the chain often, so as to preserve the gears, which can last MUCH longer.
I would do the following: buy a new chain and try. If it works, it works. If it skips, I put back the old chain and keep the new one at home. When the time arrives, I change the cassette and then install the new chain.
(Finally, some crafty people are able to remanufacture the shape of the cassette teeth by disassembling the cassette and correcting the teeth shape with a round file, one by one. I did it successfully a lot of times, but it takes some skill and a lot of time and hand labor, sort of a masochistic hobby, but anyway it works and can save you some money or at least the trip to the shop.)
How many miles do you have on the bike? How worn was the chain, when the bike shop guy used a gauge on it (which he should have done)? (Note that usually shifting problems and chain skipping are the first signs of a badly worn chain. A failed master link is not usually the first sign of wear.)
Generally a chain should be replaced about every 2000 miles (when chain stretch measures about 0.75%, and certainly no more than 1%). If you wait longer than that the sprockets "take a set" to the worn chain and a new chain will not fit as well. But if you replace the chain before it's that badly worn then you can get 5000 miles or better from a rear cluster and 10K miles from your front rings.
The "replace everything together" "rule" is stated for two reasons: 1) It makes the bike shops money, and 2) most people don't replace their chains until they're too worn.
But even then, the rear cluster is usually the seriously worn piece and the front rings are often not bad enough to need replacing. And, in any event, a competent bike shop should have a gauge for measuring sprocket wear, vs just using "the rule".