I took 1 Stick class and got my MIG certificate at a community college while getting my bachelors so I could do automotive fabrication to fuel my offroad addiction. I'm currently saving for a nice MIG welder because that was explained to me to be the best for automotive fab.
To clear things up: These are all 'arc welding' because they all use an electrical arc to melt the metals together.
"Stick welding": The official name is 'shielded metal arc welding' (SMAW).
This welding uses metal welding rods with a flux coating. As the rod melts, the flux coating creates a shielding gas that protects the molten pool from oxygen before it cools. The molten metal must cool in an non-reactive (inert) environment to ensure a quality weld. Advantages to stick welding are that it can be used outside for construction and farm work because wind isn't as much of a factor as it is with MIG and TIG, so it's more versatile. Disadvantages are that it is not as clean (more spatter) as MIG and TIG. Also, stick is a little more 'heavy duty' so it isn't as good for more delicate welding as MIG and TIG. As for metal types, stick welding is mostly used for steels.
"MIG"(metal inert gas): The official name is 'gas metal arc welding' (GMAW). This welding uses a gun where a separate shielding gas is piped in to the point of welding. The consumable metal is also spooled in from the machine. MIG is fastest type of these 3. It is cleaner than stick welding, but not quite a pretty as TIG. As for metal types, you can use mild steel wire, flux core, stainless steel. Aluminum can be used on some MIG welders with a spool gun or push-pull gun but they are costly.
"TIG" (tungsten inert gas): The official name is 'gas tungsten arc welding' (GTAW). This type of welding uses a non-consumable tungsten tip to strike the arc, and then the filler metal is fed into the already molten pool with the other hand. TIG also uses a shielding gas like MIG (100% argon I believe). TIG creates really beautiful welds and can be used on all types of metal (since you are adding the filler metal separately). You can TIG weld all types of steel, aluminium, chromoly, on and on. TIG is very, very clean. TIG also ensures good weld quality because the base metal is melted first before the filler metal is added.
As far as difficulty, they all take some work to get good at. I probably spent 5 hrs a week for a year on MIG and I'm still not as good as I want to be. For automotive work I would say MIG or TIG. I chose MIG because it's faster, and i'll pretty much only use steel so I don't need the metal choice flexibility of TIG. If you were to build a race car out of chromoly or aluminum you would need TIG, but for my the 4runner MIG is great.
As far as welder choice, I would say get the best you can afford. It would suck to cheap out on the first welder only to realize later that you need something more powerful. Also pay attention to duty cycle when selecting a welder (60% would mean you can run the welder straight for 6 minutes in a 10 minute period). If you have a low duty cycle your welder will be shutting off in the middle of your welds.
The best advice would be to take a class at a community college. The instructors have industry experience and you get a lot of time in. Good luck!