You will probably find many more openings during the contest period however, due to more people on the bands. If you are really serious about a particular contest, check the bands 28-30 days in advance of the contest. You will find that propagation often runs in cycles of that length. Some contests also require you to take breaks of given intervals. You will also learn with experience when breaks are likely to least effect your score.
Most people get their first contesting experience with Field Day.
Another good one to start with is the November Sweeps. Even a small contester with some serious persistence can work all 79 ARRL sections and earn a clean sweep cup. The CQ WW, ARRL DX and the WPX are good DX contests with lots of participation and a chance to work rare countries. The NAQP contests are held frequently and are nice for the casual contester since they are domestic contests and only last 10 hours. Before starting the contest you must decide which category to enter.