A similar application is in moving head lights - the loads and speeds are in the same ballpark. These typically use large stepper motors (maybe 2 inches square) geared down via a toothed (or "timing") belt. The gearing gets you finer movement and the ability to swing the load faster (as more torque is available) and the slight spring in the belt helps remove vibration from the stepping of the motor, making it quieter than a geared system. As a side benefit, you get to route the cabling through a hole in the large driven pulley.
In this aplication some rather cleaver trajectory planning is done, to reduce stress, achieve the shortest traverse times and give a natural looking movement. I don't know the details, but there will be some maths to do if you want that kind of behaviour.
In fact, if you could lay your hands on a scrapped moving head fixture, it would be almost what you need ready made...
The arduino idea sounds great, but is there no analog way to do this?
I'm a big fan of analogue ways of doing things (I'm normally the first to say "surely you don't *really* need a micro to do that?"), but this could be a moderately challenging design for someone who is relatively new to electronics. I don't want to put you off by any means, but it just depends how much you want to learn about electronics design, and how much you just want to get the job done.
Digital and module-based approaches seem quite popular in hobby electronics at the moment. My father, who's been in electronics most of is life, recently bought some stepper driver modules from eBay and has been pretty pleased with them; you simply set a pin for direction and pulse each time you want it to step (of course, they're actually micro-stepping). This kind of module would be very easy to hook up to a micro like the Arduino - in fact, it becomes more of a wiring job than a real electronics project. But it's a possible aproach if you're more interested in the destination than the journey.
If you wanted an analogue aproach using DC motors it would be, conceptualy at least, pretty streightforward - and would certainly make a really nice project. My aproach to this would be to place a servo potentiometer on each axis to give you absolute position feedback and control the motor velocity in proportion to the error between the current position and the desired position set by a joystick or control knob. DC motors (unless very highly geared) don't hold their position like steppers, so some kind of absolute position feedback would serve to keep the camera steady while not panning. The motor would be driven from the error signal by a hefty H bridge... not efficient, but simple.
A system I encounterd recently nested another feedback loop, using a tachomiter to measure the speed of the motor. Needless to say, it worked beautifully*.
If I were you I'd have a look around at what parts you have available, and how they fit in to your mechanical and financial constraints. Also think about what power source you want to use, as this could have a bearing on what would be suitable.