As a resident of Communication Hill (accessed by Hillsdaled Ave. entrance) my specific answers to your questions are:
University Preparatory Academy (UPA), Grades 6-12, is approx 5 minute (or less) drive away; San Jose City Community College & San Jose State University are both approx 15 minutes. SJS, located in heart of d'town SJ, is served by VTA (rapid transit);
UPA is a relatively new Charter school;with an excellent reputation. It has earned high API scores, comparable to S'toga, Los Gatos, Cupertino area achools. Has a wait-list re:students seeking entry. Located on edge of Willow Glen area, near corner of Curtne & Canoas Garden; shares campus w/Cathedral of Faith. VTA (rapid transit) access is across street.
As has been mentioned, there are many other schools in the general So. SJ area; more info re: your personal situation will enable better answers!
I would (of course) be delighted to assist you! San Francisco-born, raised in Bay Area & Santa Clara County, raised now-adult children in SCC.....am knowledgeable re: San Jose's many n'hoods (EG: Wilow Glen), as well as Los Gatos, S'toga, & Santa Cruz coastal areas!
All the Best, Sandy McGaughey-Intero Real Estate Services