Some people said to me you can take this bus across the whole city. I can understand what he was saying, but I am wonder whether I can use around here. To my understanding, what he said is to express I can get a whole sight seeing of this city on this bus, but if I use around, the meaning won't change. Therefore, I'd like to know whether the meaning actually change.
It won't change much, but there are a few context-dependent meanings.
I'm taking the bus around the city
Would tell me that you're going to get on a bus, and stop at a number of places. Maybe getting off and on the bus route at a bunch of stops, or maybe (if it's a tour bus) staying on and just looking around the area. In this case you could "go around the city" and end up at the same place you started. Often you may not even have pre-planned what places you'll stop at.
I'm taking the bus across the city
Would tell me that you've probably got a particular place to go, and you're taking a bus to get there.
I drove around Baltimore to get to D.C.
Would tell me you traveled to D.C. by taking a highway that curves around Baltimore - probably in order to avoid going through the city itself. The context here would have to be that you were travelling some distance, as opposed to touring a city.
Generally, for touring/sight-seeing, you'd use around.