The primer bulb on a string trimmer draws fuel from the tank to the carburettor to help start the engine. This small plastic bulb and attached line can get clogged or kinked. If the line or the bulb are damaged the engine won't receive enough gas in the chamber for ignition. So you should only really need to use the primer bulb for cold starts, after it's warmed up, you shouldn't need extra gas in the carburettor. If you do, you likely have a fuel line issue or a carburettor that needs to be serviced.
Inspect the primer bulb for any visual cracks or holes. Inspect the visible line for any kinks or tangles.
Locate the carburettor. On most trimmer engines, the carburettor will be near the air filter, choke and primer bulb. You may need to use the screwdriver to remove the air filter/muffler cover to access the carburettor.
Use the needle nose pliers to pull off the two fuel lines connected to the carburettor. Hold the carburettor and engine upside down and look for any drops of fuel. This will tell you if fuel is reaching the carburettor or if it's getting blocked somewhere along the way.
Drain all of the gas from the tank into a fuel container. Use the needle nosed pliers to pull out the old fuel filter and fuel line from the engine.
Replace the fuel line and fuel filter. Reconnect both the primer bulb's line and the new fuel line to the carburettor. Push the primer bulb and see if it fills with gas. Try starting the engine.
Unscrew the primer bulb from its seat. Pull out the primer bulb and the attached fuel line. Replace and reinstall new parts. Check for gas again.
Check the seals around the gas tank and engine. Make sure the gas vent is open on the gas cap. If it still won't prime take it to a professional to have your carburettor serviced.
Storing your trimmer according to the manufacturer's guidelines will reduce most gas related problems.