What kind of wax to use on birch/pine wood?
Ok, so you need easy finishing options:


Let's start with wax since you mentioned that. Paste wax is probably the easiest option, although many waxes and wax/oil blends will work. You could also try beeswax, butcher's block wax, or even paraffin. For any wax that's too firm, try mixing with mineral spirits or mineral oil and applying some elbow grease. Wax and oil provide minimal protection and are easy to put on.
Oils are similar to waxes in that they are easy and nice looking, but don't provide much protection. Boiled linseed oil is a good choice if you don't mind the yellowish color. Both waxes and oils are easy to refresh if they ever lose their luster.
Another easy choice is a wiping varnish. If you like the look of wax, these may work too. You can buy one or make your own by mixing polyurethane (glossy) and mineral spirits (25%-50% mineral spirits). Wiping varnish is easy to apply. You wipe it on, let it soak in for a minute or two, then wipe off the excess. A cotton rag is best--paper towels will leave fibers behind. A couple of thin coats will give you a nice natural look and some protection. If you want a glossy look and more protection, add a few more coats, maybe 5 total. Do a quick rub-down with 600 grit sandpaper or super-extra-fine steel wool between coats.
other oil/varnish/solvent blends are also in the same ballpark as wiping varnish. Watco danish oil is an example of one that's basically a wiping varnish with some additional oil. It goes on just like a wiping varnish, but leaves a softer finish. This means it offers more protection than straight oil, and less than a straight varnish.


If you're applying this to an assembled piece like you mentioned, something too viscous may be annoying to get into the crevices. The simplest solution to this is more thinner (mineral spirits), and fewer coats. If you're using mineral spirits, take precautions to keep the fumes out of your lungs (mask, ventilation).

There are other good choices too, like water-base or spray-can-laquer, but the ones I have listed are probably the best combination of nice looking and easy to apply for a beginner. If the heat is not extreme, I think any of these will hold up OK. I didn't mention stains or dyes because you didn't mention wanting to color the wood.

For a deep dive into this topic, I have found "Understanding Wood Finishing" by Bob Flexner to be an invaluable resource. That book, plus experience, accounts for most of what I know about finishing.