Google up the term 'Bushcraft' and see what other folks are up to. One common approach is to carry a small hatchet, a small fixed blade, and a small folder. $10 moras and $20 SAKs are probably the most commonly-carried knives by this crowd, leaving plenty of cash left over for a decent hatchet.
Get a good idea of what you really want to do out there. The better you can define you needs, the better you can meet them.
My advice? Getting hurt in the woods is something everyone should avoid. Folding saws are your friend...
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Best advise here!