Well, for Oxford you'll need three very strong references, and for Cambridge you'll need two. There really is no hard and fast answer of how to get one though- I was lucky in that I had someone that taught me for three consecutive years on something completely different to what I wanted to research, as well as having two people who both taught me for one year and then co-supervised a dissertation the following. I also had a 'special subject' which is an intensive final-year course (courses are four years in Scotland, so we have an extra year to get access to staff) which met for four-hours a week in a class of eight for 24 weeks, so my tutor in that knew me pretty well. I also had As under all of my referees, so I actually had four willing undergraduate referees who knew me well enough, which was saying something in a department of 50 staff and 800 students. That said, I also got involved in the running of the department and through the staff-student committee, as well as running tutoring for failing first years, so I got my name out there, and getting a first as well as the departmental prize helped get me the scholarship at Balliol, ultimately.
You'll get much more exposure to staff for a longer period of time in final year- in the first two we would often go through six lecturers in a 12 week term, and met for tutorials once every two weeks (so 12 times a year), and typically half of those were junior staff or PhD students- who could often be excellent teachers, although they hadn't yet got names as academics in their own right. I think again, that'll change in the next 18 months to two years. It sounds like you've made a good start to university life- but speaking from experience, I didn't win any academic prizes despite pretty good grades in years 1-3, but did in fourth year for the overall prize where it counted. I don't think getting involved within a department can harm your chances of impressing people that might be needed to write a reference though, but there's plenty of time for that. If you're heading for a first going into final year, then you'll be in a position to think about Oxford and Cambridge applications.