Originally Posted by
fietsbob
Hanging the bike by the front wheel means you don't have to lift it as far,
bike's weight heavier on the back , but ...
you have to use a hand on the frame, to lift the weight, and another on the handlebars ,
to keep the front wheel straight so as to catch the hook..
If you hang it from the back wheel you have both arms to lift the bike
and the rear wheel will not flop around in the process.

When hanging by the front wheel, I use both hands on the handlebar, saddle on my stomach and just lift it onto the hooks.
To the OP: hanging a bike by the wheels will not damage anything. Bike shops hang customers' bikes this way when brought in for service. The only chance of damage is if the hooks are too close together that someone else hanging their bike hits your bike with theirs. In this case your frame might get scratched. To mess up your gears, they would need to hit your bike pretty good.