Such a useful article here on how to choose the right white pigment. My advice? Buy the best quality Whites that you can regardless of whether you are looking for lightening or high pigmentation. Even the best are seldom that expensive. You won't regret it if you do but very likely will if you don't. It can make such a huge difference to your painting experience.
How to pick the right white pigment : Artists & Illustrators
Mixing with white can be deceptive. It is always good to remember that a colour mixed with white will look lighter when wet, and definitely on the palette than it dries. It's a tricky job getting exactly the right mix but it's also precisely the point where you need to sit down, breathe and remember you are an artist not a factory hand. Slow down, think, look very carefully at what you are doing (that's what this process is about after all!) and be prepared to experiment and discard quite a bit until your colour for a particular purpose is just right.