Go to this site to get the manual
http://www.hondapowerequipment.com/pdf/manuals/31VG4642.pdf
Sounds like your cable is broken or loose.
When you put the control in third does the cable get tight?

is there a way to install the cable without removing the rear wheel transmission

when the mower is in third the wheels will turn, but not in 1 or two

If you look under neath the selection for 1,2,3 is positive, The plastic unit clicks into position and is kinda locked. The selector from the handle has to match. So if you put the handle in the middle , which is second the position in the transmission should be in second.
If this is done first and third will be set and correct.

Special thanks to rosemallow and tomplum, thanks to your help I have gotten mower drive working again, now if I can just get my hrr216vxa working, tomplum I will try your suggestion about spraying carb cleaner and see if gaskets are okay, I think they are though they were installed new with new carberator