There are many, many options for both wood and metal finishing, so what follows is just my opinion.
Wood: Walnut or maple stain, probably 2 or 3 coats, then Birchwood Casey's Tru Oil, 3 to 6 coats, rubbing out with very fine (0000) steel wool between coats, then Birchwood Casey Stock Sheen and Conditioner, or leave it in the gloss. Tung oil is another option after the stain coats; it's more work and takes longer but is a very nice finish. I've never been happy with linseed oil, it looks good but too much work for me, but you have those young hands...
Metal: Definitely must finish the barrel. Browning is more authentic and to my eye, better looking. Two options: Birchwood Casey's Plum Brown or Laurel Mountain's Barrel Browning solution (actually there are several, these are just my favorites). I suspend the barrel using wire so I don't have to touch it with either method. And don't forget to thoroughly clean and degrease the metal before you start applying the solution; I like Blue Wonder cleaner/degreaser/rust remover, but there are plenty of good options for that. Plum Brown is applied with a cotton swab after heating the barrel with a propane torch (or baking it in an oven) until the solution sizzles a little. Heat it gradually - take your time. It will require 3 or 4 coats, although you can stop after 2 if you like the finish. The finish gets darker with each coat. Laurel Mountain is applied cold and rubbed down with a rough (like denim or canvas) cloth between coats, but should be applied and allowed to cure in a humid environment, like over a tub of water.