I can try to explain what I understand about a couple methods, mostly from news articles.

For size and distance (from their star), a lot is learned by measuring the "wobble" of the star, and fluctuations in its light output. As described in Newton's 3rd law, every force has an equal, but opposite reacting force. So, the force of gravity from the star to the exoplanet is the same size as the force of planet to the star. Since the star is so much more massive, it will barely move, but it does move. With careful observations, the resulting "wobble" can be detected, and the existence of an exoplanet or exoplanets can be deduced. This can give mass, though splitting it into the various bodies of the system would be hard. Distance also can be discovered here, though beats me how exactly. A lot of this could be done with high school physics though.

The planet's passing between the star and Earth also dims the star some. I believe this is the oldest method of discovering exoplanets. If the output of the star and the radius of the planet's orbit are know, the size of the planet's is rather easy to discover based on how dimmed the star is. How to get the radius is again, beyond me. As the exoplanet passes between the Earth and Star, it will also emit a spectrum of light (well, it's always doing this I guess, but I think it is easiest to see at this point). The composition of the spectrum tell us what elements are there, and thereby the composition of the planet.

I'm sorry, this is just what I've gleaned from news articles, but I hope it helps.