Perhaps the best small engine tip is the analogy to a campfire. You need fuel. You need air. And you need to get the fire started. In the spring, fuel is often a problem for a number of reasons, making starting a lot like trying to light wet firewood. Fuel loses it combustibility over time. Fuel stored in small gas tanks or cans is also prone to water contamination from condensation. Your best bet is to empty any old fuel that's been sitting for more than a month and start with fresh fuel. You'll probably have to premix it with two-cycle oil.
Once you have good fuel, you must ensure you have air. Air filters on weed eaters are easily soiled and plugged, as the small filters are often exposed to a lot of dirt and dust. Remove the air filter cover, remove the filter and clean it with a designated filter cleaner before replacing it. Cleaning the filter will let air flow freely to the engine.
Two-cycle engines are known to foul spark plugs. A plug is fouled when it develops a sooty buildup of carbon. Eventually, the carbon can cause the plug to stop working. It may not be necessary to replace the spark plug. You may be able to clean it. Just remove the spark plug wire, then remove the plug with a spark plug wrench or socket. Clean the electrode with solvent and a fine wire brush before replacing.
Carburetors perform the critical task of mixing the right amount of fuel and air. You shouldn't need to adjust your carburetor if your weed eater was previously running properly and the carburetor hasn't been adjusted. Weed eaters are prone to vibration, as the carburetor adjustment screws might vibrate loose or just get knocked out of tune. There are typically three adjustment screws. One is an idle screw that sets the throttle position when the trigger is off. The other two are for fuel and air. The fuel and air mixture does change somewhat with altitude and climate. If you adjust either or both of these screws, start with the factory settings recommended in your manual. Then make small, incremental adjustments of a quarter-turn of the screws. Note the adjustments so you can set it back if it doesn't improve engine responsiveness.