OK. Here's my latest version. It's designed around the way I work, the methodology I use when welding and to minimize taking pre-built modules apart. I want to be able to "fine tune" the voltage and capacitance depending on the wire sizes in use, from 36 gauge on down to 28 gauge. For example welding a 36ga wire to a 30ga wire can get a little tricky. The toughest part has been extending, physically, the trim pot on the LM2587 to allow voltage adjustment from outside the project box.
Keep in mind that this version can produce up to 60 volts at the welding connections. This is substantially higher than many other welders in this thread and could produce a potentially dangerous shock to the user. Use with caution!
As far as soldering techniques are concerned, there a lots of videos on YouTube showing how to solder. Just use a decent quality electronic solder that already contains flux. Actual construction can be literally "point to point" wiring, built on a piece of perf board or even on an etched circuit board. In terms of reading the diagram, if there is a dot where two lines, wires, cross that means there is a connection. No dot means no connection.
(to be read at double speed)
Just my opinion, worth what you're paying for it. Your results and mileage may vary. No warranty or guarantee is expressed or implied. Proceed at your own risk. The end user agrees to hold the poster harmless and without liability. I am not a doctor, scientist, chemist or physicist and hold no academic degrees, nor do I play any of them on TV.