as far as I know there is no pre-made thing you can buy that does "detect a red dot". HOWEVER, a really simple solution does present itself (although it will admittedly cost a bit of money):
Buy a microsoft Kinect, a small netbook, and buy or build a mobile robot that can be controlled over USB (Uh...COMPUTER -> ARDUINO -> ROBOT).
Using Robot Operating System and Ubuntu, and a nicely lit, white floor (it IS hard to segment images :-P) you get an image stream from the Kinect and a point cloud. the image is used to find which point in the point cloud corresponds to the red dot, compared to where you are. then just move to that point.
it's not really MUCH harder than making a mobile robot, since most of the algorithms to, say, segment out the red dot are simple enough as long as your scene is otherwise uncluttered (from your description, I get the sense that you want it to work in a simple manner first, then maybe build in some complexity). ROS has some great tutorials on how to do something almost identical.
For what it's worth, I'm doing something ALMOST the same with this exact setup: I have a Kinect that detects a blue object, uses the point cloud data to find where it is in the world, then tells a robot arm where to move. Our robot arm is a bit more expensive than a simple mobile robot, but the principles are the same and it didn't take me more than a few weeks to get the system going.
Of course...it's really hard if you're new to this
AND, Like Soeren said, you won't get anywhere near the responsiveness of a cat. But it's a great hobby platform to play around with.