This is a tough one. Now days TIG (tungsten inert gas) is probably better than any other method for welding gun parts. It's also one of the more demanding welding techniques.
Go to a welding supply house and ask for a book or a video on the subject. Read them to find out if you want to tackle it.
Simple DC semi-hobbie units are available, maybe 100 or max 200 amps for little gun work. The 200 is really overkill. If you don't intend to weld aluminum, the DC is fine,if you plan to weld aluminum you will need an AC/DC machine. Be aware you will also need a Argon gas bottle for this welding. Thats the shielding gas for the puddle, filler metal and tungsten electrode.
The down side? It takes a hell of a lot of practise to get really good at welding small parts. You also need to know the basics of heat sinks and bit of metallurgy to get satisfactory results.
The up side? It is a very very interesting skill to learn. Amazing things can be done with a Tig welder. Heat input control is unparalled, and precision of the weld can be unbelievable. IT is also a darn good looking weld when done right.