A cheap LCR meter ($25 on ebay) is very useful for checking old capacitors.
I agree with Jim that a good soldering iron is your second priority after some key hand tools. I like products from Hakko myself.
An oscilloscope is a very useful item, but probably a bit too expensive for you right now. You can do quite a bit of repair without one as many problems in old stereos are bad controls and pots, bad capacitors and bad connections. A scope isn't needed for these most of the time. Depending on your budget, you can get a brand new scope for $299 (ie. Rigol DS1102E), and used ones for less than $200.
I would prefer a simple scope over spending any money on a logic probe or other simpler devices, with one exception. In my early days I got a lot of use out of an RF/AF Signal Tracer. Mine was an old HeathKit unit which is museum stuff now, but since this type of thing is nothing more than a high gain audio amp with a high impedance input, a switchable detector diode, a speaker and meter on the output and a battery for power, you can build your own or buy a kit for very little. Google RF Signal Tracer for DIY plans. This is much better value than a scope if the funds are really tight.
At first, I think it unlikely that you need a signal generator since you can probably pick up signals off the air for checking receivers. However, in case you might consider it later on, there are many very good used ones out there that cost more than $300, but these are overkill for what you might need. Perhaps you can find on older Leader or B&K, or Instek, or something like that in the $50 range.
As Jim was suggesting, good hand tools are pretty important at the start and while you didn't ask specifically about this category, its worth your attention. You need a few basic but very good quality items including small wire cutters, tweezers and needle nose pliers. Fixing old stereos involves taking great care not to scratch panels or lose screws while you are working on them, so set up a good workspace with a soft surface on the bench, with nearby cleaning supplies like paper towels, a bottle of isopropyl alcohol (buy cheap at Costco), perhaps an alcohol dispenser, For small parts, perhaps consider a muffin pan or the like so things don't get lost. Sometimes a little magnetic pan like mechanics use can be good to hold screws. Its also worth mentioning that you need some sort of very good lamp on your bench, perhaps a simple gooseneck lamp with LED bulb or something like that, because you will need some good light to see inside the box you are working on and a flashlight just doesn't cut it. Of course you need screwdrivers and you'll probably have to invest in a set of micro-sized ones, like so-called jewelers screwdrivers or repair sets for laptops. Lots of these on ebay and elsewhere for only a couple of dollars.
Are you going to focus on cosmetic restoration at all? Perhaps you will need to learn about how to refinish a wood panel, or how to polish scratches out of a clear plastic part. At the very least, basic cleaning is required and while I like Isopropyl Alcohol (99%) for this, there are lots of detergents and household cleaners that may be useful. Removing dust is always necessary and a small vacuum cleaner is useful for this. Also, some compressed air for blowing dirt out of a chassis is really handy too. A small compressor is another big ticket item, but very handy for blowing things clean.