The rocket does all the work unless the satellite carries an inertial upper stage/apogee kick motor (usually geos going from transfer to final orbit). The small thrusters on most spacecraft are just for stationkeeping, ie keeping it in a precise orbit, and end of life operations (deorbit for LEO, acceleration to junk orbit for geo. The launch path of rockets is mostly lateral (down range), more than vertical. So it's not up then sideways, it's both at the same time. Mostly up closest to the pad, mostly downrange near 1st stage cutoff. Think more like half a parabola on its side starting at 0,0 and extending out towards positive infinity, where the x-axis is the ground.
Orbits are almost all between polar (90degrees) and equatorial (0degrees) in inclination. Inclination is the angle of the ascending node (equatorial crossing) of the ground track in relation to the equator. There are a handful of missions that have flown retrograde (greater than 90 degrees and thus aganst the rotation of earth to some extent). Rockets have the most capability (throw weight) when launched closest to the equator and in equatorial orbits due to using the earth's rotation to help with orbital velocity.
Quick orbital primer:
LEO - low earth orbit, close to earth, the ones you see at night moving very fast, ISS and shuttle fly here.
GEO - geostationary is far from earth, at the point the orbital period matches earth's rotation, so these stay in the same point in the sky all the time (think DirecTV, Sirius). Geosynchronous orbits are similar but not exactly equatorial, so they drift north and south in the sky relative to a point on earth throughout the day.
MEO - Medium earth orbit. GPS sats sit here, they are semisynchronous, meaning you see the same satellite at the same point in the sky twice a day.
HEO - highly elliptical orbit. Russia uses a lot of these, as Geos are along the equator and too close to the southern horizon in the sky for their high lattitudes. So these sats dwell a long time over a region at a really high apogee, before swinging by really close and fast and then zipping back out to appogee. This keeps them in a relatively small portion of the sky for long durations.
