This info assumes you have a 2 cycle! Your mileage may vary!
In my experience, check that the tank filter is not plugged. Make sure the fuel lines are not rotten and that the tank filter is still attached to the line in the tank. Pull the carb and buy a diaphragm/gasket kit(usually under $10) and replace. Don't think twice about re-using the old ones. Make sure you take note that the pump diaphragm(the thin one that has the two little check valve flaps) goes against the carb body and then the gasket, while on the other side, the gasket goes against the body and the rubber one with the metal disk sits on top of the gasket. Make sure the purge bulb is not cracked. Check that it's working. Remember, it's not a primer. See that it's sucking fuel up into the bulb and pushing it back into the tank. When you check the spark arrestor screen, take the time to pull the muffler and examine the exhaust port of the cylinder. That too can be full of built up carbon. I'd replace the plug and check for spark. Harbor Freight inline testers are under $10 and work fine. Lacking that, a pinch of starter fluid can tell you if have spark. If you are still no joy, I'd put a compression tester on it. I wouldn't be happy w anything under 90psi but I prefer 100+. Known good properly mixed, fresh gas goes without saying. I have to go to a marina for ethanol free so I'll use premium mixed 50:1 with synthetic oil(Echo in my case). Run it in every 2 cycle I own.
Good luck!