Garden sprayers are pretty much self-contained, and unlike the sprayers of yesteryear made of steel, brass and bronze, modern sprayers are almost exclusively made of plastic. Because of this, they are fairly inexpensive. That means that with any major problem, like the container getting holed, or the pump handle breaking, it is far more cost-effective to buy something new than to try and repair it. However, from the pump to the hoses and nozzles, there are ways to repair issues in a cost-effective manner.
Check the pump on your garden sprayer. If it pumps hard, turn it over and squirt some silicone spray lube inside. If the top of the pump is removable, unscrew it with your channel lock pliers, and coat the inside of the pump walls with Vaseline.
Check the plunger gaskets or cups when the pump is unscrewed. Grease these gaskets with petroleum jelly and make sure they are straight. Warped or damaged plunger cups either need to be replaced, or disposed of and a new sprayer purchased, whichever is more time- and cost-effective. Some pumps are permanently sealed and will not allow you to take them apart, and if that's the case and there is a problem with the pump, you'll have to buy a new one.
Unscrew the clamps at any connection that leaks with your flat-bladed screwdriver. Once the hoses and connectors are pulled apart, they can be sealed with pipe dope. Spread the dope around the fitting liberally, join the pieces together and clamp securely. If it still leaks after that, replace the O-ring at that junction. These are available at any hardware store.
Unscrew plugged nozzles with your hand and run through them with a large sewing needle. Just make sure the needle is no larger than the nozzle aperture to maintain continuity when spraying. Hold the nozzle up to a light, and if you can see through it, it is effectively unplugged and ready to be screwed back on.
Remove leaking hoses from the connector and cut them down past their leak point with a utility knife. They can either be clamped back to the original connector, or just replaced entirely. Hoses are inexpensive and available at any hardware store, and getting a longer hose at this point will increase your overall reach, and might turn out to be a bonus for hard-to-access trees and plants.