I am flying with my one year old son on my lap?

because it is a little out of my budget to get an extra seat, I will be bringing a car seat, Do you check this with your luggage to go under the plane? Or do you have to lug it around with you until you get on the plane to check it there? Anyone been in this situation before? any advice would be great. Ps. It is just me and my son on this trip (4 hour flight from WI to FL one stop in Texas)