Engines need spark, fuel, and air to run.
Check your spark plugs. Are they wet with gas? Do they spark?
Is it getting fuel to the carburetor?
Is the air filter clean and dry?
Is the choke linkage hooked to the throttle cable?
If it were my mower, I would run it level all around.
If you are going to do it by the book, match your height selection to the actual cutting height in the front and raise the rear 1/8" to 1/4".
Normally, it's best to buy in pairs.
You will get it close to the correct size by inflating to maximum pressure printed on the sidewall, and setting it out in the sun for a few hours. After mounting it on the mower, lower the pressure to the recommended psi.
That unit should take about 73 ounces of 20W50 per side. So, a 5 quart jug will be plenty. There is no need to prefill the filters. DO NOT use standard automotive filters for the transaxles.
Follow the air purge procedure described by Hydro-Gear...
I'm not a big fan of the Toro Personal Pace mowers. Sure, they cut okay. However, their cheesy drive system is their weakness. A Snapper Ninja Commercial will run rings around them.