To send a spacecraft into space?

There are theoretically several methods, but so far the only two that have actually been demonstrated to work is to strap big honking chemical rockets onto it and launch it that way or to use a "space plane" that flies like a plane in the atmosphere and like a rocket in space.

Usage of space planes has been extremely limited so far. The US Space Shuttle looks sort of like a plane, but has been informally referred to as a "flying brick" and lifts off like a rocket. The wings provide just enough lift for it to make a controlled landing, not really enough for it to actually "fly". The only really "plane-like" craft that's made it to "space" (over 100 km altitude) is SpaceShipOne, and even then it is not capable of achieving orbit and required a carrier to lift it from the ground initially.

Theoretical methods that haven't actually been used:

A balloon can be used to lift a payload, but not to put it into orbit; the spacecraft would have to carry an engine capable of accelerating it to orbital velocity.

A mass driver could be used to accelerate a craft to a velocity that could put it into orbit. This would require an extremely long track to be used for manned craft, but a shorter track and a greater acceleration could be used for unmanned payloads.

Nuclear engines (at least two different types) have been suggested, but are rather problematic. The "Orion" type uses nuclear bombs to provide propulsion (which is problematic for both ecological and political reasons). The "Nerva" type is less problematic politically, but there are still safety concerns.

Space elevators have engineering and safety problems; no known material is strong enough to support a tower high enough, and the "tethered" variety requires a material with a very high tensile strength and we don't have anything strong enough for that either. There's also the problem of what happens if it fails (read "The Barsoom Project" for a description of several types of space elevators, and the failure modes associated with each).