On my first cross-country as a student (dual), we actually transitioned through the class B airspace just north of us, flew to a smaller airport north of town, did a landing there, followed by flying back to the class B and landing at the primary airport there before returning home.
Basically, I had a few key experiences:
- The controllers work quickly and expect you to pay attention. More than once, as a rather inexperienced pilot, my instructor had to help with the radio, either in understanding the instruction, or helping reduce my confusion by handling communication.
- Even on a slower Sunday, the center was training a new controller, so there were times the person training had to step in to clear up some of the radio congestion -- we almost had to turn away from the class B, since they hadn't yet assigned us a beacon and cleared us through the bravo.
- We got vectored around pretty quickly... which, basically threw out all our flight planning (we were expecting this to happen anyway).
- Talking to approach to land, they told us to expect the visual approach to one of the parallel runways, and the tower controller didn't toss us any curve balls there. Although, with the headwinds, even with a fair amount of power, we weren't moving very quickly, and probably got cleared to land a bit too early (thankfully, the airspace wasn't busy... but there were a few jets waiting for us to land and taxi off).
- Ground control and clearance delivery was combined, and they were pleasant enough to give us progressive taxi without asking (I was expecting to write furiously), and also let us taxi to depart off the intersecting runway.
- Departing after cleared and returning home was pretty uneventful. At this point, I was out of the way for most traffic. Much more like being under regular flight following.
Granted, it's not like I was flying into ORD -- it was a pretty slow day at an airport that isn't busy much to begin with -- but I have a feeling if you've been used to flying from a Class C airport, not much will feel different for the most part, especially at one of the smaller Class B airports (and for the bigger ones especially -- DEFINITELY get to know the airport diagram beforehand).
To answer some of your specific question, the A/FD will list traffic patterns (probably going to be opposing right/left traffic for parallel runways so people don't end up head on at each other) ;) Controllers will guarantee separation from all aircraft, so you'll be assigned a heading and altitude (maybe "at or below"), and possibly a speed. So really, if your radio communications are good and you're able to fly straight and level, you won't have many problems (it just comes down to obeying clearances, and not violating any airspace). You're right that you would need an endorsement to fly solo into a Class B airspace, but this would not apply if you were dual with your instructor, since they are PIC.