Rhodes has a long tourist season by Greek standards. The island gets reliable sun from April through October, with July and August delivering Mediterranean heat in the high 30s — hot enough to make midday sightseeing in the Old Town uncomfortable. The shoulder months of May, June, and September offer the same sun with temperatures in the mid-20s and significantly fewer visitors.
October is underrated: the sea is still warm from summer, prices drop, and the Old Town quiets down enough to feel like a real place rather than a stage set. The beaches are still usable well into the month. From November through March the island is quiet to the point of closure — many restaurants and hotels shut entirely, and the ferry schedule reduces dramatically. The upside is that you can walk the Street of the Knights in solitude and see Rhodes as it is rather than as it performs for tourists.
The main festivals are Easter (celebrated throughout Greece with particular intensity), the Medieval Rhodes Festival in late August and September (outdoor performances in the Old Town), and the International Folk Dance Festival in July, one of the longest-running in Europe. If you are visiting specifically for the Old Town, any time from late September to late October gives the best combination of good weather and manageable crowds.