The Black Sea coast runs hot in July and August. The sea temperature reaches 25–26°C and the beaches are at their busiest — if you're here primarily to swim and lie in the sun, this is when to come, but expect the sea garden and waterfront to be packed with Bulgarian families and tourists from the resort towns. Book accommodation well ahead for this period.
May, June, and September are noticeably better for sightseeing. The weather is warm and dry, the Archaeological Museum is less crowded, and you can eat outside comfortably. June is particularly pleasant: the city is greening up but the summer hordes haven't arrived yet.
Winter is quiet. The resort towns to the north and south are completely shut, and Varna itself settles back into its role as a working port city. The sea garden is empty, the cafes are smaller and warmer, and a few museums go to reduced hours or close for maintenance. It's not an unpleasant time to visit if you're curious about the city without its tourist face, but transport connections thin out.
The Varna International Music Festival runs through most of June and attracts classical performers from across Europe — worth timing a trip around if you're in the region.