This may sound pretty elementary, but I'd been riding for quite a few years before I was shown the right way to get on the bike. It was part of some training that I had with a bike group that included how to stop correctly, riding in packs and getting on the bike with clip in pedals.
Whether you have clip-ins or not, the way doesn't alter.
Stand beside your bike on your favoured side. If you are right handed, thats the left side of the bike.
Turn the handlebar away from you, quite a long way, and lean the bike towards you. This lowers the top tube to almost knee height.
Place your leg over the bar and turn the pedal backwards until it is just past top center.
Clip in or just place your foot on the pedal ready to push down if you have no cleats.
Straighten up the bars and push off with the favoured leg.
If you have clip-ins the left foot is placed into the cleat on the down stroke. A little practice and its dead easy. The key to it all is turning the handlebars to lower the top tube rather than just trying to throw your leg over.
When you get mastery of your clip-ins you can start practicing pulling on the upstroke but that's another art on its own.