Ditto what Jeanette said. The typical folding saw you can find in most garden shops is much better than any wire/chain hand saw.

Also, don't forget the saw that you get with the large format (Rucksack/Trekker/Soldier) size Swiss Army Knives.

As for chopping, the Gransfors Bruks Mini-Hatchet is a very nice tool in a small (but pricey) package. Their slightly larger Wildlife Hatchet is also a good choice, but much more care is required when swinging anything sharp.