I'd be inclined to say you may have a dodgy gear selector switch. Basically it's a rig that tells the transmission to be in this gear based upon what you have selected. Eg Drive, Reverse etc etc. I doubt it's anything hugely major as for a problem like THAT the whole transmission wouldn't work. You must have park I'm assuming since you can only start an auto in park or neutral I would suggest maybe gently cycling throwing the gears top to bottom on the selector and see what happens.
I would still get it to a mechanic for a professional diagnosis but for now if it's drivable that'll let you start to save some money.
I would still get it to a mechanic for a professional diagnosis but for now if it's drivable that'll let you start to save some money.