Most garden mums bloom in late summer or during the fall. They tend to get leggy and flop all over the place, so most people cut them in half when 12" tall and then continue to pinch until July 4th. You shouldn't need to do anything else special. If you do nothing, they should bloom in about 4 to six weeks. If you let them bloom and then cut them back, you should get another flush of bloom about 3 weeks later.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.