You did not say how you are traveling, The first person mentioned air travel and is correct, but if you are traveling by bus, most buses do not have seat belts and a car seat would not be a good idea to have the kid sitting in it due to it not being restrained properly.
Yes, you can bring along your kid s stuff. If your car seat says it s FAA approved, you can (and should) bring it onto the plane so your kid can sit in it. This is both safer than just sitting in the plane seat, and your kid will be in his or her familiar car seat. If your kid is under 2 and doesn t have a separate seat, you can check the car seat. Most airlines will let you gate check the stroller, that is, you give it to them just before you get on the plane, and they give it back shortly after you get off.People travel with small children all the time. So long as you plan ahead reasonably, e.g., spare diapers for the smallest kids, picture books and games and snacks for bigger kids, you ll be fine.
You can bring the car seat on the flight with you ONLY if you paid for a seat for the child( to put them in it). If not then it is checked along with the stroller. But the stroller can be taken to the gate to be checked in there. The stroller is free I do not know about the car seat.