To reach orbit from a planetary surface, you need to do two things: attain sufficient altitude, and sufficient tangential velocity. The first point is easy, but unless you reach sufficient tangential velocity, you will simply fall back to Earth again in a ballistic trajectory. The required velocity depends on the orbit altitude and planet mass. For example, a low Earth orbit (LEO) requires a velocity of more than 7000 metres/second.
Orbital launchers usually take off vertically to clear the dense part of the atmosphere quickly, before pitching down to add to the tangential velocity component. You should always launch into a prograde orbit (towards east) to utilise the planet rotation. Orbit insertion normally occurs in two stages: the initial burn leads to a ballistic trajectory, and a second (orbit insertion) burn at the highest point of the trajectory (apoapsis) to raise the periapsis (lowest point of the orbit).
If you are new to Orbiter, you should try your first orbit insertions with one of the more powerful spacecraft, like the Delta-glider. The more realistic launchers, like the Space Shuttle, don't provide much margin for error. If you feel ready to take on the Shuttle, make sure you turn on the "limited fuel" option. With unlimited fuel, the Shuttle is too heavy to reach orbit!