What types of energy are used when someone plays a guitar?

While playing guitar, the most relevant types of energy involved in the process are the sound, kinetic, thermal, muscle, elastic and electric energy. The latter may not be involved if you're using an acoustic guitar.

The primary source of the energy comes from your body! You move your fingers and arm to play the guitar! This energy was stored in the molecules inside your muscles and now is being used to move your fingers which, in turn, transfers energy to the strings of the guitar.

Now the strings are vibrating, having kinetic energy, but also having the elastic energy (or potential energy) because the strings do stretch a little bit. What happens now depends on the type of guitar: acoustic or electric.

In the first case, as the strings vibrate, you can hear the sound - the guitar's body amplifies the propagation of these vibrations to the air! This sound comes from the vibrations in the air. So the vibrations of the strings are causing the air to vibrate at the correct frequencies. This process transfers kinetic and potential energy from the strings to the sound and thermal energy in the air (which is just potential and kinetic energy of the molecules in the sound wave).

In the second case, as the strings vibrate, a magnetic coil picks these vibrations, converting kinetic energy into electrical energy (which also is a sort of potential energy). This electric energy is then used to control a magnet which will be responsible for generating the sound waves (sound and thermal energy)!

I'll add that in all the processes, some energy is lost to thermal energy. You can't avoid this due to the second law of thermodynamics.