My Bonneville shakes violently at around/after 50mph and up while stepping on the gas pedal and accelerating. It stops when I let go of the pedal and immediately step on it again but it'll start shaking again after a while.

It also idles rough but not as bad as the shaking I get when I drive. Any ideas on what it could be?

Check and make sure all lug nuts are secure on the front wheels. Then if you can, get underneath and check for CV-shaft play.

Hmmmm...Harmonic Balancer maybe....

When did you have a good tune up done?

sounds like a balencer to me. I would also lift the front end and check for play in the wheels.
do this by lifting the car and support it. Shake the tire at 3 and 9. then shake the tire holding it at 6 and 12

The issues sound related, and would be multiplied when the torque converter is locked up, which happens at usually 48mph. You can confirm this by easing on the gas, and when the car starts shaking, if you lightly touch the brake pedal, just enough to trigger the brake switch, it will instantly get better. When the brakes are engaged, the torque converter unlocks. If you hit the brakes, and the issue remains as it did, but the rpms bump up indicating the torque converter unlocked, then the above suggestions hold true. I believe the issues are one and the same however, and it means tracking down your running rough issue. Could be ignition problems, vacuum leak, etc.

A rough idle can be caused by a dirty throttle body (or one of its sensors), vacuum leak, or ignition issue. A rough acceleration can be a problem with the ignition or fuel delivery systems. A rough idle at a constant speed above second gear can be a TCC issue, which is most often a shaking with light surging on the tach.

Everyone'* contributed sound advice thus far. As much as you don't want to hear this, it can be anything I've mentioned, anything they've mentioned, and the suspension. Start narrowing things down. Tightening lug nuts is free. Putting your front end on jackstands is free (if you have jackstands). A good tune-up isn't free, but fairly cheap.

Also, is your check engine light on? If it is, we have a huge step in the right direction.