Sounds like you have correctly checked the ignition/spark process.
So it's a fuel issue.
I'd ask myself how old the gas was. If the gas is old - more than a few months - I would consider it too old to use. So I would get the old gas out of the tank, clean the tank if I could, change the fuel filter, change the fuel pump, pull the top off the carburetor so I could clean out the float bowl,put fresh fuel in it and expect it to crank up and idle well as well as reving up smoothly.
The float bowl:
In case you are unsure, the float bowl acts just like the tank on the back of your commode. There is a float, a small valve on the input side and a teeny weeny hole somewhere in the bottom of the bowl for the fuel to get out. Blow out the teeny weeny hole with compressed air - you can buy a can of compressed air at a computer supply department.
If I was doing it, I'd buy myself some gasket material before I started so that I could make another gasket if I tore the original when I took the top off the carburetor.
If you are not too sure about your abilities and don't want to do the carburetor stuff, it'll probably be alright anyway. I just like to go through everything when I get into something like what you have.
Fuel pump:
Sooner or later it's going to fail. Change it now. Check out the old one and save it for an emergency if it pumps.