Q: IF I USE COLLEGE WIFI, WILL THEY SEE WHAT I SEARCH AT HOME USING MY WIFI, OR ONLY WHAT I SEARCH ON THEIRS?
Whoever has administrative access to the router can see what you are doing by viewing the router logs. So, the college can't see your home logs.
But that is only part of the equation. Your internet provider also keeps activity logs. However, the college would have no access to those.