Cross information

I'm a total amateur at ARCGIS, this is the first time I use it for real. The thing is I have several shapefiles with information about:

a. Schools in my city (elementary, middle and high school) b. City districts

The schools' information has Points that represents the unit (the place itself) and also a Buffer (influence's radio of every unit). ONE LAYER BUT SEPARATE SHAPES (One for point one for buffers). So, I need to cross this information so I can generate data that tells me the districts that are best equipped with each kind of educational level. I mean, which districts have more coverage of elementary schools, more coverage of middle high schools and more coverage of high schools. Even if they don't have any school within, but just some coverage from any buffer.

I don't know if I explained myself very well, but I'll be so thankful for any advice. I think I must merge the schools and then join this shape with the districts... (From what I've learned in my two days ARCGIS experience)