You are describing a car with overheating brakes. Brakes squeaking is from "glazing", which causes a high frequency slip and grip that "rings" the rotor like a bell. Wet finger on the rim of a glass is the same idea. Some squeak is normal, just annoying. The shake can be caused by a few things, but since we are talking brakes, the rotor being warped is most likely.
Why the brakes are overheating are usually due to two factors, caliper issues or improper use of the brakes.
Caliper issues can be things like a sticking piston or sticking slide pins. The technician who did the brakes should have cleaned the pins as well as the hole they go into and lubed them. The grease that the pins use to slide can gum up and cause the caliper to stick and seize, resulting in the pads not retracting away from the rotor, just like you were riding the brakes. The "saddle" that the pads sit in can also be a sticking point due to rust or improper brake pad hardware installation. The tech should have cleaned up the channels that the pads ride in and lightly lubed them. A sticking piston will have the same result, the pads riding the rotor too much and excessive heat and wear results. Too much heat cause by either issue can result in both issues being observed.
Improper use of the brakes is usually riding the brakes, like in heavy traffic, or hard braking habitually. Long braking periods of light pressure can also cause excessive heat to build.
I am not saying the shop did a bad job, but that is as likely a possibility as bad braking habits from where I am sitting with the information available.