The ~ means it doesn't matter what the opponent plays. You have mate the next move no matter what.
The T looking thing probably does stand for checkmate.
The Q/R/P followed by the mate symbol means you can checkmate with any of those pieces, possibly depending on opponents defense.
I've done some of these type of puzzles and they are hard and have little explanation. I personally don't mind this and like to figure out variations, but other might not.
Very often these type of puzzles require somewhat odd looking moves in somewhat odd looking positions. Look for moves that limit the opponent or cut off (or threaten to cut off) a key file, diagional, or square. Here is an example from a book called Canadian Chess Problems:
This position is of course winning for white, but you're supposed to find a mate in two.
Answer: (The book only has 1. R5c4 - you must figure out the rest. Of course, you're supposed to before looking at the answer. I did. Can you?)
Looking at an ebook of an old book titled "121 Chess Problems" by J.W. Abbott (1897). And in looking at the "solutions of problems" at the end of the book, I find the notation used there is like nothing I've ever seen (or could find on the internet) before. In looking at "mate in two" problems, here are some examples of the notation:
1.Rc7, ~ (Rook to c7 OK, but what does "~" for "black's response represent?)
2. Q/R/P (followed by a symbol that looks like a "T" with two "_" lines instead of one. Think this is Mate?) What do the "/" marks between Q,R,P represent?
Another example:
1.Qh6, ~ 2.Q/R/B (and a symbol-I'll call the symbol a doubled "T")
Any help would be greatly appreciated.
Thanks.