After removing the tube, check the rubber around the valve. If it is torn or cracked, replace the tube. If air is leaking from the old patch, replace the tube. Do not patch over it. Patch other kinds of leaks as shown. Complete patching kits are sold in bicycle shops. Before remounting the tire, inspect it to make sure that no glass or metal fragments remain inside.
Use a good hand or foot pump and a pocket gauge to inflate tires to the pressure listed on the sidewall. Service station gauges are notoriously inaccurate, and the volume of air the pumps give can quickly burst a bicycle tube.