I wish I was a better nympher than I am, but I'm getting there. These are my $.02, feel free to take or ignore whatever I say.
There are many ways to cast. You can cast just like a dry fly, and you can roll cast, and in certain situations those will work. I end up doing a lot of role casting actually. I found that most commonly I'm doing (or trying to do) a tuck cast, it just gets the nymph to the bottom more quickly. This is especially needed in fast, deep runs, because otherwise you're drift is pretty much over before the nymph even gets deep enough. Basically, the goal on this cast is to get your nymph to enter the water vertically and with power. I basically aim for a point 3 feet or so above the water and beyond the target, the cast straightens out in mid-air, and you stop it in mid-air, the nymph just dives and enters the water first.
A few pointers, try what you want to:
1. The closer the shot to the nymph the deeper it will go.
2. If you're not snagging up at least occasionally, you're not fishing deep enough. Add shot, move it closer to the nymph, change casts, but at least change something! A good nympher is constantly adjusting weight, leader, cast, etc. I like using weighted nymphs personally, its easier to change to a heavier or lighter fly than it is to change shot, at least for me.
3. A classic tapered leader is not necessary for most nymphing, most of the time it works against you. I go straight from my butt section to a long piece of 2x or 3x tippet, and follow it with 4x or 5x. Its very common for me to have 1 foot of butt, 7 feet of thick tippet, and maybe 1-2 feet of finer tippet. You need to open your cast, which means this is best on bigger water. But it lowers the resistance of your leader to the current, and you get a better drift with better sensitivity.
4. I've found I do better without indicators. There are exceptions, if distance casting is necessary an indicator helps, and those long slow pools of relatively constant depth.
5. When the trout fishing is slow, suckers are excellent practice.
6. 2 nymph rigs do work, but I don't recommend them until you start to do well with 1 nymph. They can just lead to tangles and frustration. If and when you decide to add a second nymph, I like the dropper method. The dropper line should be short (less than 4 inches) and a larger size (or at least stiffer material) than the line its attached to.
7. As a searching pattern between hatches, nothing beats a nymph below a dry.
8. During a hatch with rising fish, don't be afraid to fish an unweighted nymph near the surface, you might be shocked at the results.