OP, you’ll get mentorship from the moment you enroll in the university and that continues until you graduate from your program. Your mentor can hook you up with some people you’re interested in knowing or looking forward to working with. That’s part of why they’re there for you, in the first place.
I have never heard anyone complained about a poor handling of student mentorship from a regent scholar. On the contrary, i’ve occassionally heard a few of them not really being interested in seeking for advising because they’re basically contented with what they have, or they’ve already mopped out their whole college journey on their own. Or their peers and clubs (they’ve joined in) are deeply supportive of them, or they’ve already have close connection with their favorite prof, etc… Only goes to show college advising is applicable to some but not all. At Berkeley, you’ll definitely get it when you need it. It’s a myth that advising at Cal is next to impossible. It’s just that not too many students there are interested in doing it. But it’s always there.
Re research involvement, there are tons of high-quakity research being conducted in the college, more so, in the university, as a whole. Those students who show interest in a particular research are the priorities. Getting involved in research is not really the main issue for some students in CoE. The main issue is often whether they’re able to get into the research team of their choice. And this is often the same issue faced by students at many research powerhouses, i suppose including Princeton. Generally, a research team in CoE is composed of only 8 to 15 students, which often includes grad/postgrad students. But, if your prof know you as you’ve been showing interest in his/her work, chances are he/she will accommodate you even when you’ve failed to make the cut in the selection process.
The main point here is showing interest in research involvement. If you want to get involved in research, you need to show to your prof your interest in the work that he’s been doing. You need to demonstrate your capability to carry out such an important role that you’ll be handling once you’ve been selected for the team. This is true at all research powerhouse schools, including Princeton. Do not assume the profs at Princeton are the ones courting their undergad students just so they can complete their research team. That is a mistake. The only advantage I can see Princeton has over Berkeley is you’ll have lesser competition for every team. But Berkeley has got more research teams to choose from. And, i would assume the research quality is of higher standard given Berkeley CoE is a solid top 3 in America, if not the world. So, in case your first choice team is already full, there will be other teams that will be open for you.