First work out what materials you are probably to weld as there is no one answer to what is best suited.
Then work out how deep you need to go into this rabbit hole and believe me it is a wide and varied profession.
With some guidance at crucial points you can mainly self learn as I have over 50+ years after first going 50% with pop for a oxy acetylene set when I was 13 and learning the basics in engineering classes at school.
Consumable gases are your biggest outlay as bottle ownership or rental is a significant ongoing cost however a gas set is the most versatile in the small engineering shop. Cutting, heating, fusion welding, brazing, soldering, metal powder overlays and even flame hardening make an oxy set the leader for general use. With skill all metals can be welded with an oxy set. Best suited for under 5mm fusion welds. Gas bottle hire/ownership ongoing costs.
Arc is probably the next easiest to learn and least outlay but is mostly limited to 2mm+ thick steel although thinner can be successfully welded once the skill develops to manage molten weld pools and not have them too large where they drop onto the floor. Leaning to manage the arc is a skill on its own as is selecting the right welding rod for any application or steel type. Traditional arc is mostly limited to ferrous metals of most any thickness.
Mig is the next cheapest investment and successfully manages thin panel steels right through to heavy fabrication, is fast to apply and when using Mig gases there is no welding slag to remove which can be a significant advantage in post weld cleanup. However some of the nicest Mig welds I've seen were done with flux cored wire and CO2 as a shielding gas. For the home shop 0.6mm wire is adequate for most jobs but for heavier work 0.8mm wire might be preferred.
Mig's great disadvantage is susceptibility to wind that blows the shielding gas away making it troublesome to use outdoors in other than the calmest conditions. Mig shielding gas required. (Argon/CO2 mix)
Tig is the jack of all trades but not suitable for heavy work and also susceptible to wind. Tig can weld a wide range of metals however the power unit must have the specialist features in order to do so and these require a deeper investment for that versatility. The best machines and operators can weld the thinnest of metals with Tig such is the control you can get with it. An auxiliary treadle control is highly recommended. Argon shielding gas is required.
Of the 4 types mentioned I have used all but Tig the least as its weaknesses are a liability to my needs when I already have Mig, arc and oxy. Actually I do have a Tig torch that fits a little 140A DC caddy welder I rarely use but the oxy is more convenient. I only have the caddy welder for the occasional portability needs.
Other stuff.
Auto darkening helmet, just so cheap these days you're are nuts not to have them.
If you eyesight is aging, and for who of us isn't it, use specs when welding. Even $2 pairs help immensely to produce better quality welds.
Grinders, a small angle grinder at least for weld prep and post weld cleanup. A cupo brush for it is a good investment too as are cut off wheels.
Safety glasses, you'll need them when grinding and removing slag.
Files, why ? There are materials that can be contaminated with carborundum from a grinder and a file is required for the last stages of cleanup.
Gloves, yes there are times when you really need them with Mig in particular and it's splatter can make the process of laying down a nice weld most uncomfortable. They'll also prevent numerous burns while you learn your craft.
Slag hammer, sorta goes without saying for arc and gasless Mig.
Wire brushes.
Cramps, several will be required as will backplanes, jigs and straight edges to cramp the work to to keep it straight while being welded.
Squares, protractors etc for markout and setup.
Pitfalls
Metal identification is crucial to successful serviceable welds.
Preweld prep is also crucial to successful serviceable welds.
Some cast metals require careful pre-weld preheat and post weld cooling to prevent cracking.
There is lots to learn, enjoy the ride.