Do autopilots have an 'activate' switch/button, in addition to 'on' and 'off'?
It is complicated (-;

First thing first, autopilot systems come with a variety of forms and complexity. The autopilot on a Cessna 172 or Piper Cherokee is going to be very different than the one on a Airbus or Boeing. Different manufacturer will also design their autopilot interface differently; while there are certainly similarities, there is no set standard in the aviation industry for autopilot interface and behavior.


  (...) is simply changing the altitude and heading numbers, or (...)
  there is some sort of button that commands the autopilot to execute
  the new commands?


Both.



Here I will discuss using the Boeing 777 autopilot, which I am familiar with. On the autopilot panel, there is a heading window, which displays a 3-digit number. Directly below the window is a knob, turning the knob counterclockwise will decrease the heading, while turning clockwise will increase it. The knob can also be pushed.



The heading section of the Model Control Panel on a Boeing 777. The knob is a 5-in-1 control: the outer ring can be rotated, the knob can be rotated, and the center of the knob is a push button.

If the autopilot is in "HDL SEL" mode, it will fly the heading specified in the window. In this mode, if the pilot rotates the knob to adjust the heading, the new heading will take effect immediately. However, if it is in "HDL HOLD" mode, turning the knob will adjust the number, but it will have no effect - the plane will hold the current heading. Pushing the knob will change the mode from "HDL HOLD" to "HDL SEL", and the plane will then turn towards the selected heading.

A similar logic exists when climbing using the "VNAV" mode. Let's say the plane is initially cleared to 5000 feet after takeoff. The pilot set the altitude window to 5000 prior to the takeoff roll. Once autopilot is engaged, it will climb to 5000 feet, after which it will hold that altitude. At any point, the altitude knob can be rotated to set a new altitude, say 7000 feet, however it will not take effect - the plane will still climb to 5000 feet, then stop the climb there. To climb to a higher altitude, the pilots have to push the knob, after which the plane will initiate a new climb from 5000 feet to 7000 feet. This is because ATC often clear planes to "step climb", i.e. climbing then holding at certain altitudes until reaching the cruise altitude. The autopilot function is designed around that.




  where is that switch (or button)?


It depends on the model. Other manufacturers call them differently, but the logic is similar. Also, advanced autopilots have a lot of functions; the ones I discussed are just selected examples about heading and altitude selections. Pilots can also "arm" an autopilot mode, i.e. mark it so that it will become engaged when certain conditions meet. The specifics of that are too long to describe in this post.

Like I said, it gets complicated.