He is BRILLIANT for tacking up - stands nicely, doesn't fidget, and drops his head into the bridle for you (but has to be reminded to take the bit, which doesn't bother me) but once I go to mount, he steps backwards to follow me and doesn't stand still til I've sorted him out three or four times, or someone is there to hold him for me.
Take his head and bend it towards you. If he still doesn't stop then take him and turn him in a circle until he stops on his own. Keep repeating this until he realizes that it's easier to stand still.
After I'm up, he's eager to be off and going and will walk off. That's pretty bothersome, as now he's walked off when I'm only half up and I've had to fling myself over the saddle and squirm upright while he's cheerfully carrying on his way. He hates standing still, and will fidget, paw and act like a twit tossing his head until we're going again.
Make sure when you're mounting you have his head turned towards the side your mounting on. At first make him stop until all four feet stop, don't let him walk forward. But here's the hard part to remember! When he stops completely, as soon as you feel him get ready to walk off put your legs on him. So it turns into YOUR idea, not his. Keep doing this until he realizes it's easier to stand still. Plus make your horse have a real "Workout" while riding. It will make everything easier.
And then.. There is his weaving. And some other things. When we go out alone (we typically go with my friends and do leisurely walks), he weaves constantly and is completely dead-sided to leg when I KNOW he knows what I'm asking, because he's perfect when my friend is there. And he balks if I correct him too much for his liking and we dance sideways.
This could be anything really. But from what I've read here maybe your over correcting?
I know I can handle his surprise!Balk/Spooks, but the abrupt sidepassing sideways that he'll pull when he doesn't want to go forward without caring what is beside us really worries me. One day, he didn't want to cross a road with a ditch/creek on either side (road is all earth and dirt, completely and utterly solid and very wide) when he crosses it all the time fine and he abruptly went sideways into some downed treelimbs and didn't seem fussed at all.
First walk him beside it a few times, then once you get the confidence to go over it make him walk over it. Even if it takes an hour, make him go over it. Don't ever let him refuse. It doesn't matter how he goes over it, the first time is never pretty.
Don't worry about what your friend can do. She probably just rides with more confidence. I hope you enroll in lessons they make horse owning so much more better. And your horse has learned that he can get away with things with you.