Zagreb runs opposite to the rest of Croatia. Its peak is December, not August, because Croatian tourism goes to the coast in summer and Zagreb empties.
By season
Spring (March–May). April and May are excellent — the Green Horseshoe in leaf, the Botanical Garden open, terraces properly outdoors, low prices and a city working normally. May is arguably the best month of the year.
Summer (June–August). June is warm and long-evening. July and August are hot — regularly into the mid-thirties — and half of Zagreb is at the sea. Restaurants close for a fortnight, the theatre season has ended, and accommodation is cheap. It is quiet and it is a city with the lights half on. Zagreb Summer Evenings, classical concerts in Upper Town courtyards, are the best thing in August.
Autumn (September–November). September is the other excellent month. October is fine and greying. 1 November at Mirogoj — candles on every grave, the whole city there — is extraordinary and impractical. November is the dullest month.
Winter (December–February). Advent from late November to early January doubles central hotel rates and fills the centre. Cold, dark by four, and lit. January and February are the cheapest, emptiest weeks, and the city is entirely itself.
Fog is a real feature of the Sava plain in winter and can sit for days.
The one decision
If you are coming for Advent, book months ahead and go on a weekday.
If you are not, avoid December entirely. It is the only month that materially changes what a trip costs.
Festivals, roughly
April Zagreb Dox. May Subversive Festival, Design Week, Cest is d’Best. June INmusic at Jarun, Animafest. July–August Zagreb Summer Evenings. October Zagreb Film Festival. Late November–January Advent.
Dates move; check before you fix anything.
The longer version is Zagreb through the year.