Building a multirotor "from scratch" is a daunting prospect. I would recommend that you first buy a kit from a a company such as Hobby King.
But, If you are going to design and build your own machine from scratch, you will need some tools and supplies. The list can be intimidating, but you would be surprised how most of the stuff listed below can be used for a lot of other household and hobby tasks as well. Although you can get by with less than the extensive list below,having everything listed will make your job a lot easier and save time waiting for deliveries from China.
Two soldering irons (or a temperature-controlled one with interchangeable tips). One should be high-power with a large tip (for 10-12GA power leads). The other should be 35W or so, with a fine tip, suitable for PCB work.
Solder
Two wire cutters (dikes). One should be small for cutting 18-30Ga wire, the other should be large enough to easily handle 10GA.
Small needle-nose pliers. Most of the ones you find will be too big. I like the ones that are about 4" long overall.
3mm socket head capscrews of all lengths. 25 of each size - 5mm (thread length), 8mm, 10mm, 15mm, 25mm, 30mm, 40mm.
Lots and lots of 3mm nylok hex nuts
7/32" nut-driver (works fine with 3mm nuts)
2.5mm hex wrenches - buy 10. You will use them for nearly EVERYTHING.
Small metric hex wrench set (so that you will have all the sizes that aren't 2.5mm).
English hex wrench set with .050 as the smallest size (believe it or not, some Chinese equipment is built with English-sized setscrews).
3/8" and 1/2" aluminum angle and channel stock (Home Depot or McMaster-Carr).
Hacksaw with fine teeth.
Dremel tool with assorted abrasives/cutters.
Electric drill, preferably one that can be mounted in one of those cheap drill-press stands.
One of those cheap drill-press stands.
Small center punch
XActo knife
Metric drill set. (or a very complete English set). Make certain that the smallest bit is no larger than 2mm (.079")
Some very limp 12GA and 20GA wire in red and black (silicone insulation)
3.5mm "bullet" connectors, male and female.
2 pin JST connectors is both genders.
MT60 connectors in both genders.
Shrink tubing in all sizes.
Heat gun (air).
Hot-melt glue gun.
Digital calipers.
1/4" Tapered reamer
Voltmeter. A cheap one ($10) will do.
Lots of small tie-wraps
Double-sided foam tape
Adjustable square.
Some .050" or .060" aluminum sheet
Some .063" bare fiberglass sheet (bare PCB material).
A 3D printer is not necessary but very useful
Patience!
It should be noted that I have everything listed above except the last item, and I'm still hurting for a few tools, like a sheet metal shear and a metal bending brake. I can borrow those for the time being, but I'll eventually need to buy my own.
I'm just going to have to learn the patience part on my own.
.