Whatever bus you can get, change at Rotherham Interchange and get the train to Sheffield.
From there you can either take the #60 (Destination Fulwood) from outside the train station forecourt to West Street area for the Management and Engineering buildings, or behind the Students Union for the rest of the University; or take the tram (Destination Malin Bridge), to University of Sheffield stop.
As for passes; that allow this sort of multi-operator, multi-modal journies... you'll be looking for a pass from the South Yorkshire Passenger Transport Executive.
You'll be crossing from the Rotherham 'Zone' to the Sheffield 'Zone' in terms of tickets, so you'll need something that covers all South Yorkshire.
There's a "Day Tripper" (if you start your journey before 9.30AM) which costs £5.50 for all modes; or £4.50 if after 9.30AM.
Now, for longer term stuff, and Dependant on when your 19th Birthday is....
Up until your 19th Birthday you're entitled to a Monthly Travel Master costing £39.50. This allows unlimited travel on Trains, Trams and Buses in South Yorkshire at any time; so even your local buses when you're back home at the weekends or at night.
After your 19th Birthday that doubles to £75 per month; so £18.75 per week.
There are Week passes available at £20 a week, so if you know it's the holidays and you're not going to be at Uni for 3 out of 4 weeks of a month.
Check out TravelSouthYorkshire.co.uk for more details.
You'd be able to buy the Day Tickets from your local bus driver at the start of the day; the longer term stuff you'd have to buy from the Interchange building at Rotherham or Sheffield.