The main things are a) the type of roller used b) your strokes c) having enough paint on your roller d) usually it is safe to assume 2 coats are needed
It's important to have the right kind of roller for the type of texture you are covering. For example, if the ceiling has a textured finish, you will need a 'hairier' roller that will cover all the dimples/bumps etc. more easily. Using a smooth roller over a textured finish means the paint won't penetrate the gaps as much so that's important.
It's all in your strokes. The longer the strokes, the better. However, if your ceiling is big and it's difficult for you to do it all in one stroke, you can split it in two and do half strokes instead. It's best to start your stroke away from the edge of the previous one (as the first stroke always contains more paint) and work it into the previous stroke. There are videos on youtube that demonstrate a good stroke. It's worth watching as many people just roll it on any which way and then you end up with stroke lines all over the place.
Don't be afraid to put enough paint on your roller. If it's dripping all over the place, you have too much though.
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It should be heavy, but not dripping all over and you should be able to do a good bit in one go.
Generally, when painting, a general rule is to expect 2 coats. The first is the main coverage, whereas the second perfects it and makes it look better. You really shouldn't have much problems with covering white, you just need to take it slowly and make sure you have enough paint on there covering it. Your second coat should fix it.
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