I would go for something that can take small sized animals for food. I wouldn't buy something sized for deer myself. .22LR, .22mag or .17hmr would be good choices. Personally, I'd opt for the .22LR. I would also have to recommend something that will reliably shoot when dirty or abused. That means no semi-autos. And preferable no external (removeable) mags. Stick with lever, bolt, or single shot - they are not ammo sensitive and will feed anything. Get tube fed if it's a multi-shot. I wouldn't want the bulk/weight of a scope either. A tang sight would be ideal - just about as accurate as a scope at distances you'd be shooting with a .22LR. If a tang sight wouldn't fit your rifle, then a receiver mount peep sight is a good second choice.

Picking one rifle to do it all, my choice would be a Henry lever action in .22lr with a tang sight mounted. I have the Henry H001T and I love it, but I would not recommend it for your trek. Simply because it has a heavy octogon barrel and you wouldn't want to lug around that extra weight. Go for the basic normal-barrel Henry (the H001), or the carbine (the H001L), or even the youth model (the H001Y). However, the heavy barrel octogon model (the H001T) has a barrel already dovetailed to allow multiple front sight options so that's a plus, but the downside is it's extra weight. A Lyman globe sight up front would be ideal for your adventure. And a Marbles or Lyman rear tang sight. The Marbles rear is adjustable for windage, the Lyman is not (but it's cheaper).

If you don't like lever actions (Who doesn't?! They're great!) you could go for the Marlin 981T bolt action tube fed .22LR. It is noticeably longer and heavier than the Henry though. The 981T is a nice .22LR rifle, but the Henry is much faster handling.

I don't have any single shot .22's in my collection, so I can't recommend one. But they are so simple in operation that I can't imagine you'd go wrong with any one you decided to buy.