From my observations there are three major technical things people do wrong when popping up:
1. hand position - the purpose of the arm thrust (push) is to get your hips high enough to bring your legs under. If your hands are too far forward, then you cant get your hips into the right position - they will be too low and there wont be enough room for your legs. Make sure your hands are no higher than shoulder level and preferably at rib level. A useful tip is (for a natural footer) is to have your left hand around mid rib height (just below shoulder) and right hand close to the bottom of your ribs. This gives you a natural 'twist' as you push up. Remember that most of the time, at the 'top' of the arm thrust, your shoulders should be no higher than your hips. When you are on the board moving from flat to sitting (at the line up), push yourself up to sitting by putting your hands in the correct pop up position and it will become natural to place them in that spot.
2. trying to push your upper body to be over your feet, rather than bringing your feet under your upper body. Essentially your push up should move your upper body straight up from the board and then your feet come underneath. A lot of people seem to want to plant their feet straight away because they feel strange and so push their upper body backwards to be over their feet. This is almost impossible to do and, even if you manage it, you are in the wrong place and heading backwards. This problem seems to occur a lot on steeper waves when people freak out because their board is pointing downhill and the natural instinct is to try and make the board level again by planting your feet toward the rear of the board.
3. trying to stand up straight too quickly. After your feet land, if you try to stand up straight too quickly then your shoulders are heading up and backwards and your feet (on the board) are heading down the wave and forwards. So chances are you will fall off. Keep low - keep your hips low rather than your shoulders (bend at the knees rather than the waist). You can stand up after a second or two, when you are stable.
The other major issue, which is not a 'technical' issue, is simply not committing to the pop up. Doing a half assed push because you are a bit scared or not confident. Surfing is a matter of committing - you want a wave, commit to it, paddle hard and pop up hard. Paddle softly and you perl, pop us softly and you fall. Its like riding a bike - try to to it slowly and its harder than doing it faster.
Home practice is good, but a better way is to find somewhere with a lot of white water. Catch the white water and pop up. Repeat 50 times. Come back tomorrow, repeat 50 times. There is no way you will get 50 practice pop ups on unbroken waves in a session but you can on white water. Eventually you will peg the pop up in white water and can move out back. It is a bit different out back, but there is less difference between white water pop ups and unbroken waves than there is between floor pop ups and unbroken waves. There is an embarrassment factor in catching the white water, I know, but if you have a choice between constantly wiping out or playing in the white water for a week and then being able to actually surf, just harden up and take it. No actual surfer cares if someone is at white water level, we all were at some stage. Its like a learner driver, its a stage you go through. Just keep out of the way.
Another tip is to find a surf school in session and watch people. You very quickly see who is going to perl and who is going to miss their pop ups, and you can usually see why those occur. So don't do what they are doing.
As to which way you pop up - whichever feels right. I'm a left footer (kicker) but pop up natural (left foot forward). Actually, for most people, their strongest (most stable) leg is the one they dont kick with - its the leg that is used to hitting the ground and keeping the body in place while the glamour leg hits the ball. Your preferred leg makes no difference to your pop up or which foot should go forward, although it may make you a 'front foot' surfer or a 'back foot' surfer once you are up and cruising.
Welcome to the large school of people who have learnt that snowboarding and surfboarding are not quite the same...