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Most satellites are launched to a mere fraction of the altitude at which geosynchronous satellites orbit.
How are geosynchronous satellites carried so high? Do they store a lot of propellant on board? Or do they keep their final stage longer than low-altitude satellites?
Altitude is part of the problem. Assuming you wanted to get to GEO from a circular, 700 km altitude orbit you'd first need to do a burn of about 2.24 km/sec along the orbital velocity direction to raise the apogee of the transfer orbit (the half-ellipse):
The spacecraft wouldn't normally do it itself, the launcher upper stage would typically inject you into a Geostationary Transfer Orbit (GTO) directly.
What every spacecraft has to do itself, however, is circularisation of the orbit once it reaches the geostationary altitude. This is done by doing yet another burn along the orbital velocity direction, but this time of 1.2 km/sec. You can think of this burn as trying to raise perigee altitude and circularising the orbit or matching the spacecraft's velocity to match the geostationary orbit's velocity. Any inclination changes are also done there to reduce the mass of the fuel necessary (the slower you go the less fuel you need to change inclination by some amount).
And that's how you get to GEO. So the problem is, by the measure of delta-V, mainly due to altitude, less so regarding the velocity. The only problem is every spacecraft has to do the circularisation burn itself, which adds some complexity.
Satellites are launched to a range of heights from about 160-1200 km (LEO) out to 33800 km (geosynchronous).
Altitude is not the challenge here. Orbital speed is the hard bit. From sciencelearn:
even though it takes nearly a million joules of energy to lift a 1 kg mass to an altitude of 100 km, it takes over 30 million joules of extra energy to give it enough speed to stay in orbit around the Earth.
Your last two sentences are correct though - the further out you want to go, the more propellant you need to carry, and you'll keep your stages burning longer. I know this sounds very simplistic, but it really all comes down this in the end: delta-V.
Have a read of the other questions tagged launch for useful information on speeds, launch types etc.
As an example, Ariane 5 (large PDF: Ariane 5 user manual) generally delivers its payload in a Geostationary Transfer Orbit (an elliptical orbit with its highest point at geostationary altitude). From there, the satellite's own engines circularize the orbit.
It depends a bit on the rocket: the Ariane 5 ECA has a second stage that cannot be restarted, so it cannot do the circularization burn. With a restartable second stage, it's possible to push the satellite into its final orbit, but then you have to do another maneuver to remove the rocket stage from geostationary orbit.
