Q:

Why are rockets launched vertically?

Why the rockets mostly used for space missions are launched vertically upwards and not horizontally like an airplane carrier, gradually increasing its altitude, wouldn't that be more efficient?

A:

To launch an object into a stable low-earth orbit requires accelerating the object to orbital velocity (approximately 5 miles per second) tangential to the Earth's surface.
Intuitively, getting out of (the majority of) the atmosphere and then accelerating more or less tangential to the surface seems like a good plan. This implies a nearly vertical ascent early and then a gradual gravity turn to a nearly tangential trajectory. 
A hybrid combination of a large airplane-like vehicle that raises the rocket high into the atmosphere and then launching it from there is another approach which is used for smaller launch vehicles, e.g., the Pegasus launch system.
At least one company, Stratolaunch Systems, is taking this approach seriously for a larger launch vehicle.

