It's actually pretty straightforward:
1.If the minecart is in your inventory, drop it. If it is already on the ground, proceed to step 2.
2.Press "u" when you are 1 tile away from the minecart (or standing on the same tile as the minecart).
3.You will be presented with the following options:
4.If you "Push" the minecart, if will move a few tiles in the direction you chose. Physics comes into play here, so it will gain/lose speed depending on the usual factors.
5.If you "Ride" the minecart, you will hop into the minecart, even if you were a tile away, and it will move in the chosen direction with you in it. It will gain/lose speed depending on the usual factors. Whilst the minecart is in motion, you should press "." (period) to skip your turn; if you attempt to move whilst the minecart is still in motion, the laws of physics come into play, and you will take damage. Alternatively, you can push the minecart whilst it's still in motion (although I'm not quite sure how one can bend physics so as to push a moving minecart whilst the person is inside the minecart). If you push it in the same direction you were travelling, you will greatly increase the minecart's velocity. You can also push it in different directions, and this will cause it to gradually change direction-the amount of pushes this requires depends on the minecart's velocity. Once the minecart has stopped moving, you may move out of it safely, or you may want to give it another push. Note that if you "Push" a minecart right after having ridden it (still on the same tile as the minecart), it will act as though you chose to "ride" it.
And that's basically it.
Tip:If you want to test this out yourself, arena mode allows you to spawn minecarts.