Why don't space rockets use wings like airplanes, but overcome gravity by rocket thrust alone, as they rise vertically? Wouldn't wings provide extra lifting power?

"Wouldn't wings provide extra lifting power?" Yes they would, but mainly at low speeds. Once the speed of sound is passed, lifting power ("lift") drops and air resistance ("drag") rises steeply, and it soon outweighs any advantage a wing provides. To reach low Earth orbit takes a velocity 24 times that of sound, so overall, a wing raises more problems than it solves.

One way of taking advantage of wings is to use an airplane to raise the rocket above the densest atmosphere before it is fired, avoiding wasteful air resistance. The "Pegasus" rocket is launched that way, from a specially adapted L1011 airliner (a B-52 bomber was also used). The first stage of that rocket also uses short triangular wings.

Ordinary rockets rise vertically, gradually tilting over into the direction of the orbit: the vertical ascent gets them out of the dense atmosphere in the shortest time, reducing losses due to air resistance. One possible way for a future space launcher to get more boost per gallon of fuel is to "breathe" atmospheric oxygen, in place of the liquid oxygen carried aboard nowadays. "Hypersonic ramjets" are still in the experimental stage, but engineers hope to create a design that saves fuel at up to 6-7 times the speed of sound, a reasonable speed for a first-stage spaceship. Like the Pegasus, it might be released by an airplane, continue to fly in the high atmosphere at the appropriate air density, and it may use short wings, or more likely, replace them with a flat "lifting body" providing enough lift at such high speeds.

A forerunner of such a system was recently unveiled by Burt Rutan, the engineer who also designed the "Voyager" airplane which circled the globe nonstop and now hangs in the Smithsonian in Washington. Rutan seems incapable of designing an airplane which is conventional, nor one which isn't strikingly beautiful, and his twin-jet "White Knight" is no exception.

Its elevated cabin straddles "SpaceShipOne", a 3-passenger spaceship with stubby wings of which parts hinge upwards to act as air brakes on reentry. It is meant to be released at 50,000 feet and to fly by rocket power to 100 km, then glide back to Earth. A ride might ultimately cost as little a $100,000 per passenger, rewarding riders with a view from near space, 3.5 minutes of weightlessness, and reentry stresses peaking briefly at 5 gravities. For Rutan it may earn a $10,000,000 prize offered for the first craft capable of such performance.