Wood will do just fine. I've killed bears, lot's of deer and small game with wood arrows. I've also been pleased with carbon arrows.
Key points to consider:
1) Make sure the arrows are matched in spine and pretty close in grain weight.
2) Make sure they're made from the best shafts you can afford (if for hunting big game) if making your own, or from one of the recommended arrow makers (like Arrow Art, Elite Arrows, ect. of which many reside here in the sponsors section of Tradgang).
3) Learn how to straighten wood arrows so that you can keep them in top shooting condition (this is not tough, so don't let it scare you off)
4) Make sure you understand broadhead mounting onto wood arrows. 99.9% of the time, when you screw one of the modern screw in points like a Muzzy or Razorcap into an aluminum or carbon arrow, it'll spin true without any issues. When I mount broadheads onto a wood taper, it may take a little tweaking to get it to spin true. The better your tapering system, the less likely that there will be issues.
Good wood arrows are a joy to shoot and one of the pleasures of traditional archery in my opinion. Learn a few of the key tricks and you'll enjoy them as well.