I was planning to reply earlier but forgot, and it took a while to find this thread again. This is actually a very wide question, which probably explains why nobody has touched it yet. I'll try to start 
Let's start with the equipment. A soldering iron is of course essential. It doesn't need to be latest high tech, but for anything more delicate than speaker capacitors, it should have a thermostat. Otherwise it gets soon too hot and you might destroy the circuit boards, as the copper tracks may come off from the PCB when overheated. Small components are easily destroyed with too much heat as well. At home I have a more than 25 years old Weller TCP (the kind where you adjust the temp by swapping the tip) and it still gives great service.
As for hand tools, you'll get nowhere without side cutting pliers (small, the kind that cuts flush on the surface) and needle nose pliers. Of course some good screwdrivers so you don't have to destroy any screws, and for B&O from the '90s and later, some Torx drivers (T15, T20, T25). Maybe even some wood work tools for the real vintage
The next thing you will need is a multimeter. Don't buy the most expensive Fluke right away since there's a good chance you will destroy at least one meter while learning. Even the cheap looking Chinese digital multimeters that cost 10 euros are nowadays quite useable for basic measurements. Later, they will also work as a backup for the occasions you need more than one meter at a time. The one thing you might want to upgrade are the measuring leads, and get some very small clips for them. You'll be surprised how often you would need at least three hands to keep the tips in place while turning a trimpot.
Very soon you will find yourself having to desolder components that have more than two legs. There are of course vacuum desoldering guns built for just that, but you can manage surprisingly well with just some good ol' desoldering wick. If you know a component is faulty, cut its legs (you had those very thin cutting pliers, remember?) and take them out one by one. A manual desoldering pump will be helpful as well and they are cheap.
My choice for the next measuring device - even ahead of an oscilloscope - would be an ESR meter. Almost every time you fix some 10+ year old B&O you will be looking for bad caps. This device will make it a little bit easier than just replacing everything in sight and hoping that was the problem.
Then we get to the oscilloscope, but in reality you might never need one. Using one to good advantage will require a lot of knowledge and experience as well. Then on, signal generators, FM test equipment, ... you can fill a house and spend a fortune if you want.
And then the theoretical knowledge and practical skills - money won't get you that. An evening school would probably be a good idea if you have a possibility and the time, otherwise you will just have to start working and learn in the process. Go to a library and see what books they might have.
One complication is that many repairs we talk about here are about old age problems, and you don't usually find much about that in text books - it takes a lot of experience you can only get by working on real problems. It can get to be like detective work. Service manuals never discuss them either!
Learn to work methodologically and think before acting. Make notes,
keep parts in good order, take some pictures while you go. You may
think you'll remember where everything came from, but after something
keeps you away from the project for a day or two, you'll have no idea!
And don't start with the shiny Beomaster 7000 or something like that as your first project! You'll get good advice on the forum, but there are plenty of ways to mess seemingly simple things up when you have no hands-on experience. There's always going to be some steps that a pro takes for granted and doesn't remember to warn about. Start on easy simple devices (and problems), perhaps buy some silly construction kits, build them and try to understand how they work.
Good luck - and when in doubt, ask before ruining something! Beoworld, while being a niché forum if I may say so, is also very fortunate in that we have people who know these products inside out, and don't keep it all to themselves. If you don't get help here for a problem, there's a good chance you shouldn't try it at home anyway