USC is the standard setter in our conference and, conveniently, we're in the same division as them and so we play every year.
Over the long term, that is where our rivalry hatred will flow. Year-in, year-out they are the team on our schedule that is most likely to be ranked (unfortunately, due to the fact that a ****ty 7-5 ND team will usually be ranked, we're not the team on USC's schedule most likely to be ranked - but we are the second most likely). More years than not, we will have to beat them in order to play in the P12 championship game - and that will be true for them as well, more years than not, they'll have to beat us to get to a championship.
Utah is going to be a secondary rival, they're the last game we play each season, so championships and bowl game positioning will always be on the line in that game.
To have a real rivalry: you have to play often, and the outcome of the game has to frequently have a bigger impact than just "another win." How much are we going to hate UU if they keep us out of the P12 championship game this year? How much will they hate us when we keep them out? How much will that USC loss piss us off if they tie our conference record and win the south?
1. USC
2. UU
And conveniently, they both wear red.