Why won't the mower move forward or backward even though the engine is running and the blade will engage?
First off, riding mowers do not "go into gear"; nor can they be shifted into or out of gear...nor, as a matter of fact are they "automatic" or do they have "automatic" transmissions. Instead, the so-called hydrostatic "transmission" is nothing more than the sense one gets--encouraged by mfr/seller misdirection (or product ignorance)--that a mower works just like a car...rather than being powered by nothing more than a belt that turns a wee little hydraulic motor, the speed and direction of which is controlled by the foot pedal controlling volume of fluid coursing though it to hydraulically drive the drive wheels.
If your rider will not move, that means one of two things (other than fatal failure mode)...or rather two of one thing, the latter being that hydraulic fluid it not flowing, is not pumping freely, through that motor.
Reason 1 (the more likely) is that the manual-mower-push switch on the motor has been engaged to permit the mower to be pushed without the vehicle (drive wheels) being blocked...because the engine is not running, so the fluid cannot flow with the turning of the drive wheels (unless the switch is engaged).
Think of it this way. When the mower is running and the clutched engine PTO is driving the belt or shaft that drives the hydraulic motor, the direction is chosen by reversing direction of fluid flow that the motor is circulating under pressure (also called pumping) to the drive wheels. Push toe and it flows and drives wheels in the mower-forward-drive direction; push heel and it flows the other way. Now, what about when neither is pushed? Then, essentially, all channels of flow are blocked--"neutral" (if you want to think of it that way) is not freewheeling (as with a car), it is drive wheel blocked by hydraulic lock at the motor. That is why a "hydo..." equipped mower does not coast as does a car. Alternatively, suppose that you could lift up and try to rotate the drive wheels in a particular direction. You would find that the wheels resist turning (unlike rotating RWD car drive wheels to remove a drive shaft, for example)...because they are essentially hydraulically locked by the motor that is not allowing compensatory fluid circulation in any direction!
Now, to disable the manual push mode that allows the drive wheels to rotate for manual pushing, look under the mower at the drive motor. It will be at the end of the belt loop or shaft being driven at the PTO beneath the engine. Now, find some kind of protruding lever big enough to grasp. If it seems "up" push it down; if "down" push it up...just make it go the other way to release. Now, start up and drive the mower normally. But wait! Before you mow, read the below.
Reason number 2 for unexpected drive wheel drive failure is absence of fluid to drive them....because you have not check fluid level and maintained it as the instructions directed you to do. So find the fluid plenum/level-check container and add some fluid if it is low. Just a little. Then start the engine and check lever with motor running. Fill carefully to indicated level but be very careful not to over fill. The mower should now run with valuable maintenance lesson learned. And remember that neglected fluid check can greatly shorten hydraulic motor life. Oh, make sure to check fluid level again when finished....after any air has been purged...add if/as necessary.