I think my tire is losing air
Tire leaks are usually traceable to three different causes:
Inspect the tire for anything unusual looking. You're looking for anything that isn't tire colored and doesn't brush right off. Nail heads can look like rocks, but they don't brush away. If you're in a quiet environment, and the leak is a fast one, listen to the tire as you spin it around to see if you can hear any hissing. If you find a nail and the tire is still holding pressure, DO NOT remove it on the side of the road. Get to somewhere safe like a gas station, then find help. You can carry a plug kit with you for this situation.
If you have a nail hole in your tire, no shop will patch it for liability reasons. Patching can be done and is safe only if done well, and if the hole is in the middle of the tread. Sidewall holes cannot be safely patched, and the tire must be replaced. It's always safer to replace the tire with a new one than patch it, and this is the recommended action if you're uncertain of what to do.
You can check to see if the valve stem is leaking by mixing dish soap and water about 1:10 (more water than soap), and applying it with a brush or spray bottle around the valve stem, including inside the stem. If any bubbles form, that's most likely where the leak is. Valves can be tightened with a valve stem core tool, available at any automotive store. They're usually either built into a valve stem cap, or come as a separate tool. This will solve internal valve stem leaks. It's unusual, but sometimes a valve stem core may need to be replaced -- these are inexpensive from the automotive store and easily done by anyone who can turn a screwdriver. If it's leaking around the base, you need to get the tire off and install a new stem.
Using the soap mixture from the previous paragraph, apply it around the bead of the tire to check for leaks there. This is the part where the rubber of the tire meets the metal of the wheel rim. Again, look for bubbles. If you see any, the tire needs to be reseated.
Finally, if you've got a sufficiently large container, you can fill it with enough water to entirely cover the lowest part of the tire and rim (ie, less than 1/3 the total height of the wheel and tire) and run the tire through that. Look carefully for a small trail of bubbles and it will lead you right to the leak. If you're only losing a couple of PSI a week, you probably won't be able to see the leak in a water tank.
If you decide to take the tire to a dealership or tire store, be sure you know what the problem is and that you can show it to the mechanic. Otherwise, you'll have to pay them to find the leak.
If you decide to patch your tire instead of replacing it, there's only 1 acceptable patch, and that's a mushroom style plug/patch that's applied from the inside (the tire has to be removed from the rim). It's only to be done if the hole is within the center 1/3rd of the tread, and you lose 1 speed rating for each patch. This is only for the rear. A front tire should not be patched under any circumstances. When patches fail, you have a SUDDEN loss of air, very much like a blowout. It's FAR worse than getting a nail, or any other type of slow leak.
Bike shops usually won't patch a tire, but normal tire shops might fix it for you if you carry in just the tire (off the wheel).