What can I use on a bare wood sign to protect it from the elements?.

I would use marine polyurethane, it is made to be outside in all elements. We used it on our back steps when we remade them and they are in the midst of their third winter and look the same as when we stained and used the marine polyurethane on them

Either brush-on polyurethane or a good tung or mineral oil. With oil, you'll have to reapply every 6 months or so.

There are a lot of sites that talk about protecting this type of project, like https://www.popularwoodworking.com/projects/outdoor-finishes, and the cost may be high. But, what you can do is go on a site like Freecycle.org and see if anyone has some left over. It's a really neat site, and everything is free:) You can join any groups, that's free too, and ask for or give things away. The only rule is that it has to be free:) You can also check with Trashnothing.com. Also a free site. Or, you can check one of the home depot type stores and see if they have small cans or cans that are dented and you can get them cheaply. Good luck with this.

marine grade poly finish

look for a poly that is specifically for outdoor use.

Polyurethane works best.

We have always used Thompson's Water Seal for any wood that is outdoors.