The task is perfectly feasible; its done all the time in the commercial satellite world (to the extent that on an outside broadcast truck with a satellite dish on the roof, you just say which satellite you want to use, and the control system moves the dish to point in the right direction).
For the hardware side you do indeed need two motors; one to rotate the dish (i.e. set its 'azimuth') and one to tilt it (set its 'elevation'). Stepper motors are easy to control; its also possible to use ordinary d.c. motors. You'll fairly definitely need position feedback in some way, if only to be confident that you're pointing in roughly the right direction.
To know whether you have the optimum dish direction, you need to measure the strength of a signal coming down from the satellite. Professional systems measure a fixed signal, called the 'beacon', which is permanently transmitted by the satellite. Sometimes identification information is encoded onto this signal. You'll most likely need a special 'beacon receiver' to accept this signal. At the other end of the spectrum are the cheap signal level meters which measure the total level of the signal being received; you could adapt one of these so that you can read the level via an A-D converter. Alternatively it may be possible to extract useful information out of your satellite receiver.
For the control side, the options are basically simple or clever! On the simple side, you just manually move the dish around while observing signal strength. On the clever side, having roughly positioned the dish you move it slowly around in a search pattern. Typically you need to cover 0.5 to 1.0 degrees in each direction. While doing this you note increases and decreases in the signal strength, from which you can determine and go to the optimum position. This optimisation can be a continous process; some satellites move around a bit in a predictable way, so the dish is continuously moving.
For a normal domestic dish, the positioning shouldn't be too critical, since they have a fairly wide angle of view. (A large antenna, such as 'Arthur' at Goonhilly, has an angle of view which will be less than 1 degree.
These links will give you some information about satellites generally: