Here is the update:
Rebuild kit completed on the carb, everything looks good and nothing is sticking or worn.
When I push start it turns over, and over, and over, but no fire. The spacer between the jug and carb "breathes" each time it turns over. It swells up, which makes me think there is too much air coming in. Problem is, I have tried the pilot screw in a number of different positions but that doesn't change anything.
I checked both valves and valve clearance, the timing chain was off one link but I fixed that too. Occasionally if I sit and jiggle the throttle it will act like it wants to start, and on a rare occasion it actually will for a few minutes.
I'm at a loss, and I really don't want to take it to a shop, so does anyone have any suggestions? Keep in mind I am pretty green, so keep the lingo to a minimum please!