King and rook versus king is a must-know mate. Unlike the queen, the rook needs more help from its king.
Use the rook to cut the enemy king off along a rank or file, restricting it to a shrinking region of the board.
Bring your king up to face the enemy king with one square between them (the opposition). When the kings face off, a rook check on the far side forces the enemy king back one rank.
Alternate gaining the opposition and checking to push the king to the back rank, then deliver mate with the rook while your king holds the opposition. Watch for stalemate when the king is in the corner.
If the rook is ever attacked, simply move it far away along its line — it keeps the cut-off.