The short answer is spring to mid-summer. When the weather begins to get warmer the cholla, prickly pear and some other cacti start blooming. This can be April or even late March in the case of an early spring or mid to late April in years when cooler weather persists. Saguaro cacti normally begin blooming in mid-May through June although an early spring can have some start blooming earlier in May. Barrell cacti tend to start blooming in July. If you are planning a trip to the Southern Arizona desert your best bet is probably the last couple of weeks in May through mid-June which is when you are likely to encounter the greatest number of cacti in bloom. There are numerous species of cacti as well increasing numbers of hybrid verities. Depending upon the type of cacti the flowers range from small and rather common to very large and spectacular. The Arizona-Sonora Desert Museum and the Tucson Botanical l Gardens both have a large section of various species of cacti and are a great place to see many types in one place. Both have websites where you can check on current blooming times for their cacti and other plants. Of course, not all will be in bloom at the same time as different types bloom at different times starting in Spring through mid Summer.