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.
Most contests have qrp, low power and high power categories.
Make sure to note whether the break point between low and high power is 100 or 150 watts. Many contests also have the option of single or all band entries. If you have a great antenna system dedicated to one band but weak antennas on the others, this may be a good option for you. If you use a packet cluster and would like to check packet spots during the contest, this will put you in the single op assisted category. Using a packet cluster in a few contests put you in the multi-operator class, though, so make sure you check the rules before using the packet cluster.