It really depends on just how far into Bonsai you want to go as to what tools you will need. For just the odd tree or two, Liatris has a couple of good ideas - a bent fork as a root raker and a cheap pair of nail pincers ground to sharp edges for a basic branch cutter. My root raker is a bent phillips screwdriver and I've had that one for 20 years now. I don't have one of the expensive shop bought ones. The shop bought ones have two or three tines on them so they may be quicker to work with, but my bent screwdriver does everything I want it to.
The other necessity is a chop-stick. It can be wood or plastic, it really doesn't matter. The chopstick is for working the soil in round the roots when you re-pot and is a necessary part of re-potting. If you don't work the soil in around the roots, you leave large air pockets and any roots in those air pockets cannot draw nutrients, so they die off. The other thing it does is to help make the tree more secure in the pot.
There are Japanese and Chinese tools available. The Japanese ones can be very expensive depending on the manufacturer but they will last you a life-time. The Chinese tools are fairly cheap but don't have the quality of steel in them, so depending on how many trees you intend to have, they can wear out after only a few years. You can also get stainless steel ones, but personally, I prefer the black carbon steel tools because they hold a better edge.
An example of price differences is that in the photo below, the medium branch cutters (No.9) are somewhere around $75.00 for a reasonable Japanese pair, and the Chinese equivalent ones are between $16.00 and $20.00 depending where you get them. On that photo, No's 1, 4, 5 and 8 to 14 are all Japanese tools, the others are Chinese.
In the second photo are 'non-bonsai' tools that I also use: Pliers for straightening out wire afer it has been taken off a branch (too expensive to just throw away), Normal garden secateurs (these are an old pair of wilkinson sword that I've had for many years). Small secateurs, they cost me $2.50 from the 'Warehouse' - don't know if you have any Warehouse stores in WA. A knife. Another heavier pocket knife. a $2.00 pair of 'garden scissors' also from the Warehouse, and last is a diamond sharpener.
The two angle grinders in the photo above, one with an Arbortech cutter on it, the other with a mini-arbortech attached. Both of these are used for taking off a lot of wood quickly and for carving dead wood etc. Next is my bent screwdriver 'root raker'. I've had it so long now, I don't even remember where I got it from. All I did was clamp it in a vice and bend it over.
My small branch cutters (No. 10 in the first photo and above) are the first pair of cutters I bought 20 years ago. They are Japanese and at that time cost me somewhere round about $48.00. That is the branch cutter I use more than any of the others. I've used them so much and sharpened them so many times that I've had to grind into the handle (blue arrow) to allow the stop (red arrow) to slot into it so that the edges of the blades come together. Eventually, I'll have to cut the bottoms of the handles off otherwise they will end up touching as well.
Below is an explanation of all the tools on that first photo:
1. Heavy scissors for pruning and trimming roots - they come in a few different sizes.
2. Light scissors for trimming foliage and removing individual leaves - they come is a few different sizes.
3. Small scissors for light trimming of roots.
4. Straight tweezers for pulling out roots or removing caterpillars etc.
5. Bent tweezers. Both of these have a 'spatula' on the end of the handle for levelling soil etc.
6. My custom made root raker. Two and three tine rakers are available from stockists.
7. Chop-stick for working soil in and around the roots - can be wood or plastic.
8. Large branch cutters. They are ok but I'd prefer to use a pair of secateurs to cut a branch off then clean it up with the medium cutters and 'knob cutters'.
9. Medium branch cutters. This is the size most people have.
10. Small branch cutters. The pair I personally use the most. All branch cutters make a concave cut.
11. Medium Knob cutters, used after a branch has been cut off. It creates a 'ball shaped' concave wound and allows the tree to callous over level with the surrounding bark. If you just cut a branch off flush with the trunk, when the tree callouses over it, it will leave a bump on the tree.
12. Small Knob cutters.
13. Branch splitter. Specialized tool for splitting a thick branch to make it easier to bend without the branch snapping - seldom used.
14. Wire cutters. These are designed to go right up to the branch to cut through wire without marking the bark on the branch. You can use ordinary wire cutters but they tend to come to a bit more of a point and if you go right up to the bark to cut wire, there is a good possibility you may damage the bark. Not too much of a problem on rough barked trees, but on a smooth bark tree, the mark will remain evident for quite a while.
That isn't a comprehensive list, there are many more specialized tools like Jinning pliers, designed to crush and remove bark where driftwood is wanted, carving and shaping chisels etc, but it might give you an idea of what you can look forward to.
For two or three years, I just had my small branch cutters, those white handled Secateurs, cheap pair of scissors, penknife, bent screwdriver and a chopstick - and I did everything I needed to with those.
Oh, one other thing; I don't think you can make a concave cut with secateurs.
If you have any question don't hestitate to ask on the forum