There's this image of a gunsmith standing over a milling machine or lathe making a complex part.
The truth is, most of a gunsmiths time is spent sitting at a bench using screwdrivers, punches, and fine stones working on a small assembly or part.

To that end, you need a good bench light that can be positioned where you need it. A good office light will do fine. You can't have too much light.

You need to be able to see what you're doing so a good magnifier visor is necessary.
Don't cheat on this, buy a good one like the Donigon Opti-Visor.
To determine what focus you need, just position a part at a comfortable distance and measure to your eye and buy that focus length.
You can also buy different focus lenses for the visor and accessory jeweler's loupes that attach.

Have a comfortable seat, and cover the fine work area of your bench in some material that's easy on your eyes.
I had a bench covered with pale Green non-shine Formica.

Have several good vises.
You'll need a big one for big projects and several small ones for holding small parts.
The more adjustable they are the better.
A good standard for a bigger vise is the Multi-Vise. There's also copies made in China.
Brownell's sell the good grade and a Chinese version.....

BROWNELLS MULTI-VISE™ | Brownells

For small parts, I found a surprisingly good Chinese vise sold by Harbor Freight.
It needs a few alterations, but is a very good small tool.........

Table Swivel Vise

Forget the small "Pana-Vise" types that mount with a vacuum bottom to the bench.
These seldom stay put and they're made of aluminum.
These are okay for model railroading but just don't have it for gunsmithing.

An excellent, very high quality vise for doing delicate trigger or small parts work is the expensive Forster "Swiv-O-Ling" vise.........

Swiv-O-Ling Vise - Forster Products

Small tools like jewelers tweezers, pliers, and Swiss needle files can be bought from online watchmaker and jeweler supply house like Otto Frei and Jules Borel.

Last, don't buy a lot of tooling until you know you'll actually need it.
Like most craftsmen, you'll wind up making some of your own specialty tools.