I think you're underestimating how sharp an edge you can achieve with stone! Early handaxes were indeed used for cutting (e.g. to butcher animals), and by the middle stone age (Mesolithic/Epipalaeolithic) people were producing "microlith" blades which had an extremely sharp cutting edge. So much so that a friend of mine carries one instead of a penknife – it's much easier to get through airports!

But before about 10,000 years ago substantial structures were rare (Mezhirich is one spectacular exception) because people lived very mobile lives, and investing time and effort in building wasn't usually worth it. So while they would have used smaller sticks and branches, felling a tree for lumber wouldn't have been common. For that we have to look to the later stone age, or Neolithic, when people began to clear areas of woodland for farming. Polished stone axes, which will have been hafted, became ubiquitous in the Neolithic and were probably used to cut down trees. They were used until recently for the very same purpose in places like the New Guinea Highlands. In the Neolithic, stone axes would have been used to clear land for farming and to build houses (the average Neolithic house in central Europe, for example, needed a lot of wood!), and were valued highly enough to be commonly placed in (male) burials.

As for further reading... prehistorians tend to get very involved in stone tools and I wouldn't recommend most of the literature to my worst enemy. Is there a particular period you're interested in? The "Stone Age" was a very long period spanning from the first humans to the first civilisations, so it'd help if you could narrow that down (and I'm afraid the "pre stone age" doesn't exist!).

This is a photo (click on the pic for a close up view) of a redwood snag that was in the process of being felled in the traditional Yurok fashion when it was abandoned. This photo is from Yurok territory on the Klamath River in Northern California. It is approximately 4 feet in diameter (based on the length of the bracken fern sword fern fronds at its base. Redwoods were felled by first burning them then cutting parallel lines, across the cut, utilizing adzes with beaver teeth or sharpened California Mussel (Mytilus edulis) shell blades. Then the wood between the cuts (splints) was split out using antler wedges. This was followed by building another fire and repeating until the tree fell. It was pretty dicey business. The photo is from the Humboldt State University collection. The description I have condensed from Kroeber's Handbook of California Indians 1970, Stephen Powers' Tribes of California 1877 and P.E. Goddard's Life and Culture of the Hupa 1903.