Hi and welcome to MLMF. Hope your Thanksgiving has been a good one.
I think you're pretty lucky to have someone give you such a "junker." From what I've heard and read, Honda made some pretty advanced riders. I think yours has a completely enclosed automotive type clutch and transmission.
One of the first questions that always comes to mind to me when someone is giving, or selling a mower, is, "what made it a junker in the first place?" What issues did it have that the owner decided to give it away. Were there any transmission or clutch issues? Was there a problem with the engine?
From what you write, you must have the 5 speed gear driven version. But if it happens to be the Hydrostatic version, there is something that has to be disengaged to allow you to push the mower without the engine running.
How long has it been sitting? First, looking over the owner's, I see it has a drum type brake, and there is a parking brake lever down on the right hand side. I'd start with seeing that the brake is not set, and that it's releasing as it should, both at the parking lever and the brake pedal. Can it be pushed when the clutch pedal is depressed enough to disengage the clutch.
Now, I'm not familiar with their riders, but I do know that the drive wheels on all of the walk behinds I've had anything to do with will not turn when you pull the mower backwards if the transmission is engaged and the engine is not running. I don't read anything in the H3011 owner's manual that says it won't move if not running. It does say that something about the clutch is automatically disengaged when moving between gears, and engaged when placed into a gear.
I believe owner's manuals are a must. if you don't have one, you can download it at the Honda Power Equipment web site. I believe shop manuals are a necessity to anyone working on their own equipment, but they can be pricey, so they do require a commitment to the machine.
Hope that helps some. Let us know how things go, and post a picture or two. We're a pretty visual group her.