1) The choice of flies would depend on the area you fish. Local shops may have the answer for you. For the most part though just be observant when you go. Look at the type of flies around. Determine whether their caddis or mayflies. Then take a look at the size and color of the fly. Once you know this you can match it with what you have. This is matching the hatch. Now some can give you the scientific name (don't worry about that stuff) and all that but really I just look and match to what I have too what is hatching. I have made flies from what I have observed in my area and if I am out fishing and see something else that I don't have I remember it and when I get back home I tie up some for next time I am out. Maybe you don't tie and if so try to find something similar at your local shop. I will say this though in time you might just start making your own flies.
Some will use an attractor. These can look like a mayfly or caddis but do not represent any particular one, only difference is maybe they have odd colors. These provoke the fish into biting because the fish is usually curious about what that fly is, or they are just acting aggressively towards it. Nymphs would require that you do a little under rock exploring. For the most part if you have an idea of what is hatching just use a nymph similar to that.
2)In a 20' wide stream I would not worry too much about waders unless there is a lot of cover around shore and you need to get in the water to cast. Waders are great but I will give one point of advice on this and that is make sure you know how to read the water that way you can plan how to approach the stream. If you just start wading in anywhere you can spook the fish and they may become weary. I have seen many people wade right into a pool without checking in close first and then they wonder why they aren't catching anything. There are some good sites on reading water but the one I liked the most I cannot find so try these out.
The goal is to fish the water before wading into that spot thus decreasing the risk of spooking the fish. Mind you sometimes you have a good idea that there won't be any fish in a spot but you never know.
3) As for not catching on a nymph I hear ya. I am working on this skill. The fish may not have liked the nymph you chose - wrong species, size or color. There is some good advice that can be given on here about nymph fishing but I am not that great of a source. I am putting all the knowledge I have learned into practice. You have to be very observant and strikes can be subtle so use a dry fly or any sort of indicator (everyone has their own preference) to help in detecting strikes. There is lots of info online on how to set this up.
4) This may relate to why you are not catching on a nymph and that is you are not getting the nymph down deep enough. As Kerry said you need some weight to get the nymph down especially in current. Try beadheads and there are also split shot methods for this, but again I am not that great at fishing nymphs so check that out on here as well.
I hope this helps. I would have gone into more detail but that would be a very long post. If you need help with any of these details I can help as best I can. There are a lot of people on here that can help as well. At times you may think there is a lot to this fly fishing but really just take it easy and don't get to overwhelmed with all the techniques. Sorry if any of this is not new.
Welcome to the wonderful world of fly fishing, you will have a hard time fishing bait after this.