If a radio tunes to a specific frequency, where does the excess energy go? If one continues to hit the resonant frequency, shouldn't the wire begin to melt at some point from too much energy?
You have a few different but related questions so I will try to explain them in a simple, no-math way.
Almost every object that has radio waves (electromagnetic waves) around it absorbs some of the radio energy. When the radio waves hit the electrons in the atoms and transfers a bit of momentum and energy to the electrons. If the material absorbing the radio waves is an insulator then most of this energy is turned into heat. If the material is a conductor (like an antenna) then a very small electric current is created. The shape, size, and length of an object has some effect on which frequencies it absorbs best so this is why different types of radio antennas are different shapes and sizes.
When a radio is designed to tune a specific frequency, it is still absorbing many other frequencies. Those other frequencies just get filtered out, meaning that energy is turned into heat.
The current created by the frequency being tuned is amplified with a bunch of transistors (formerly vacuum tubes) and turned into a strong usable signal.
You are right that if an object is absorbing a lot of radio energy it will heat up. For normal, every-day settings though the amount of radio energy around us is very low. Objects do heat up a tiny bit but this heat gets transfered to the surrounding air and the objects stay cool.
If the antenna was in a vacuum with nothing around it for it to transfer heat to, then it would get warmer and warmer. The antenna would get warm until the amount of energy it radiates via black-body radiation matches the amount of radio energy it is absorbing.
There are cases where so much radio energy is being absorbed the objects can get very hot. This is how a microwave oven works. The microwave puts out many watts of radio energy which gets absorbed by the objects in the microwave and they heat up because of the huge amount of energy they're absorbing is much faster than they can get rid of the energy to the surroundings.
I think you mean how is a radio able to transmit and receive at the same time. That's a complicated question but most simple radios can't. They're either transmitting or receiving and they must stop one to do the other. One somewhat easy way to transmit and receive at the same time is to use different frequencies for send and receive.
The simplest way to put information in a wave is to use "amplitude modulation" (AM). Another simple way is with "frequency modulation" (FM). There is a huge branch of math and physics behind the more complicated methods.
When radio waves hit the antenna it creates an electric potential difference between the antenna and ground. An electric current flows from antenna to ground, through the radio receiver. The radio receiver is able to extract information (the signal) from this current and amplifies it. Virtually all the electromagnetic energy collected by the antenna flows to ground, both the signal and non-signal components. The currents are small and there is too little resistance heating to melt wire.