First, usually it is not a move which gives check and not a move which takes a piece, because this is considered to be not elegant in modern studies. For example 1.Qd6+, 1.Qc7+, 1.Bxc5+, 1.Re6+, check with a knight on d5, c8 or a4, these are probably not the solutions. Ignore these moves and only come back to them when you did not find the solution using the following method:
Step 1 (this step can also be left away in mate in 2 studies): Check what would happen if it was black's move. Which moves would lead to a white mate in one? From this you can learn the weak spots of blacks pieces. E.g. here you can learn that a queen move which loses control over b2 may lead to 2.Rb2 mate. A rook move away from the sixth rank may lead to 2.Re6 or 2.Qd6 mate. His bishop is pinned, if he moves it, this leads to 2.Bxb or 2.Qxf2 mate. If he moves his rook to c6 this leads to 2.Ned5 mate, if he moves his queen to b5 it leads to 2.Ncd5 mate.
Step 2 Act as if you had two moves in a row. Which maneuvers would lead to mate? This would be 1.Kc8 and 2.Qb8 mate, 1.Qh1(g2) and 2.Nc8 mate, 1.Re5 and 2.Bxc5 mate, Qe5(g1) and 2.Bxc5 mate.
Step 3 Go through the list of maneuvers, found in step 2 and consider all black answers to the first move of the maneuver. The 'bad' black moves from step 1 may give you indicators which maneuver leads to success. Here we would find that the following attempts do not work: 1.Kc8 Rc6+!, 1.Qh1(g2) Qe4! 2.Rb2+ Qb4, 1.Re5 or Qe5 Qb4!.
This leads to the conclusion that 1.Qg1! has to be the move and by checking all possible black answers we can find that each leads to a mate in one, see the lines given in Brian Towers' answer and in RossMillikan's comments. Especially the fact that the Qg1 attacks the Rg6 makes the difference, 1...Qb4 now leads to 2.Qxg6 mate.
ChessBase also has a good article which gives further hints: How to solve chess problems.