I take a spare tube or 2, a mini-pump, tire levers and a couple of CO2 cylinders. Remove the wheel and remove the tire and tube completely from the rim. CAREFULLY inspect the inside and outside of the tire for the cause of the puncture. Sometimes it's obvious but usually it's a tiny sliver of glass or a wire that's not easy to find. Find it. Or them. I've seen too many riders rush or ignore this step only to have another flat a quarter of a mile down the road.
Once you are sure there is nothing left to cause another puncture, reinstall one side of the tire on the rim, lightly inflate the new tube by mouth or a couple of pump strokes to round it out, install it in the tire and reseat the other tire bead. Check to be sure the tire is evenly seated and the tube is fully inside. Inflate fully using the mini-pump or CO2. I use my mini-pump to start the inflation to be sure the tire is seated properly and the tube holds pressure and then bring it up to full pressure with the CO2.