A bus transfer is just a little piece of paper you get if you need to stop and catch another bus because the first one you're on doesn't go where you need to. Like, say the bus that goes by your house goes to the next town, but you need to only go a few miles, then you want to go in a different direction than that bus goes. If there's another bus that goes where you need it to, you can get off near a place where that next bus will be stopping, and catch that, using the transfer. If I recall correctly, the transfer saves you money. It's been a few years, though. Since the bus you're taking goes where you need it to, you won't have to worry about a bus transfer. If you get a bus schedule, you should be able to find info on transfers, if you need one in the future.

As for carrying stuff with you, I generally just plunked it down in an empty seat next to me, if available, and you can also put it in your lap. If I remember right, there might be a space above where you can put it, too.