30 years of fixing/upgrading/restoring bikes has taught me one thing - get quality 1st time around. This does not mean you need to buy workshop quality tools all the time, which can be very expensive, but spending more money on some key items will stand you in good stead for a long time and make each job easier.
For example I used to use a cheapish crank extractor which could be a pain to use sometimes. Cost around £5. I later on bought a Park Tools one which was triple that price but I found it was better as going into damaged threads and pulled cranks so much easier - it appeared the threads where a lot stronger.
Cone spanners are another good example - had some cheap ones at first but they were rubbish. Bought the Park pro quality ones and they are light years ahead in quality and ease of use. You don't need to get a whole set at once, maybe just buy a couple of sizes (2 of each) which fit Shimano hubs which are the most common then add more as you need them.
Some stuff isn't worth buying the cycling brands e.g. hex keys (buy a Wera set), screwdrivers, spanners (get a
Halfords Pro set, cheap and amazing guarantee).
Pedros also make great tools too, you don't need to stay with one brand.
If you have nothing now I'd get a decent allen key set - you'll be using them a lot - Wera here for £25
https://www.toolstation.com/shop/Au...Wera+Ball+End+Multi+Colour+Hex+Key+Set/p71306
Park tool crank puller £13
https://www.chainreactioncycles.com/park-tool-crank-puller-ccp-22/rp-prod34314
Shimano cassette removal tool £5
https://www.chainreactioncycles.com/x-tools-cassette-lockring-tool/rp-prod10183
(removal tools for freewheels are many and varied, this is the most commonly used, just buy more when you need them)
Any chain whip will do as they don't take much force through them but keep them oiled as my last one rusted away as I didn't look after it!
Adjustable spanner - I have an old Bahco one which is amazing as it has no slop.
Cable cutters - I have some old Bahco side cutters which I used for years then I bought a Park cable cutter which I thought would be better - it isn't!! Lesson learnt.
Installing/removing headsets. I don't have a special installation tool - I went to my local bolt stockist anb bought some threaded bar and big nuts/washers which works well for the amount of times I need to use it. I'd like a proper one but they are very expensive. I did buy a headset cup removal tool for about £10 which was miles better than using copper pipe or a screwdriver and hammer.
Hammer - get a normal hammer and a dead blow soft faced hammer which are ace for bike stuff (google what they are!)
..... and so on.......
If you do a job on a bike and you need another tool get a quality one to complete the job. Compared to the parts you have to buy and the time you spend doing the job its only a small additional cost - especially if you compare it to getting a shop to do it!
Last note - restoring bikes means you have to cope with stuck seatposts, bottom brackets which mean you often need very good quality tools as you have to use a lot more force through them. You'll have to learn tips and tricks to help you there - one example when I restored an old MTB was that it had a cup and cone BB and the fixed cup on the RHS only had 2 flats on it which needed a headset spanner to undo it. It slipped as it was rusted in so I used some threaded bar and big washers to clamp the tool onto the cup then used a scaffold pole for more leverage - very brutal but it worked! Main lesson is that don't go swinging at things and going medieval on stuff - use your head and physics to help you!