Yep, my beginner soldering recommendation is the Haxxo FX-601 I mentioned - it's a good iron, small and portable, and will last a long time.
The supplies (braid, flux, and solder) are all things you will need. The other stuff solves specific problems, and is the type of thing you accumulate over time.
this (PDF warning) is a good beginners soldering guide, but focusses on working with circuit boards - more common pinball soldering will be wires to lugs or other wires, which is very simple.
If you have a local hacker or maker space, they can be good places to learn this kind of thing - it's really not a hard skill, even kids pick it up quickly.