<p>Santa Clara, UCSB, and UCSC are good for CS too.
(I’m assuming you’re in-state and have parents with deep pockets who’ll fund your studies.)
In LA itself or nearby, you have LMU, Chapman, Whittier, plus the Claremont (HarveyMudd would be tops but I don’t know if they take transfers and it’s probably harder to transfer in than to UCLA; however if UCLA is a back up, then HarveyMudd may be within the realm of possibilities.)
The difference between public and private tends to be in the environment: privates tend to have smaller class sizes, closer and better advising (advisers you can “drop by” to see vs. upon appointment, personal adviser vs. general adviser, etc.), more funding for research presentations/conferences, material that’s more up to date (or, when broken, that is changed more quickly). This may or may not matter to you.
Then, you have the university’s academic level: classes with stronger students can go faster, more in-depth, vs. classes with weaker students. This has nothing to do with private vs. public (CSULA and UCLA are both public but the class content and expectations are different).
Size also plays a role: a large university will offer more classes. Note, however, than you can only take 5 classes a semester.
ABET-accredited programs are similar in content and curriculum for the science classes.
Finally, I’m a little concerned about the way you’re talking to adults who volunteer their time to help you. </p>