We have two bathrooms, back to back, and when we flush the toilet or run the shower in #2 the water level in the toilet bowl and tank in # 1 rises and falls several times. You can hear, and I've seen, the fill valve turning on and off in tank #2. Any idea what the cause of this is? Could the vent for #2 be blocked? Thank You in advance for any and all help.

All those fixture share a common vent and there could be some vent blockage. Try rodding the vent from the roof if its safe to get up there. I don't know how old you home is and who did the work. Some builders put too many fixtures too close together on the same stack and that would be costly to replace.

Bal addressed the water level in the bowl raising and falling correctly by pointing at the vent. Let my address the "bounce" of water in the toilet tank which is a different story entirely.
This is caused by a minor version of water hammer. The "bounce" in your tank should not have happened if your air chambers, (shock arresters) weren't loaded up with water and needs to be recharged, (more about that later). Any cold water valve will make the float ball bounce. If you don't believe me just pull the lid and open and close the cold water on the bathroom sink and watch that sucker bounce.
Back in the 60's we were required to to put shock absorbers,(air chambers) on all bath lavatorys and kitchen sinks. Over time water and condensate built up in them and they must be recharged. First turn off the house at the main house shut off valve. Then open hot and cold faucets at the farthest bathroom to prevent air lock.(This is important.) You will find under each fixture little brass or chrome valves. These are called angle stops. With a small pail held under the angle stop to catch the run off look under the handle of the stop and you will see a nut with the stem running through it. This is called the bonnet packing nut. Put your wrench on it and back it off counterclockwise. Now open the stop and remove the stem and washer assembly, The stop will begain to drain. When it has quit draining reassemble the stop and move on to the next one. Do this on each one of your angle stops. There will be six of them. When you are done, close off the faucets you have opened, turn on the water to the house, and give yourself a pat on the back.
You have just recharged your air chambers and saved yourself a hefty service call bill. Hope this helps TOM