Since Budapest is smack in the center of the Pannonian Plain (yes, even Europe has steppes) expect hot and dry weather well into fall. This is particularly troublesome in Budapest, where humidity also is higher. Later in the season the Continental climate comes into play, days get colder and frost emerges by early November. Only some snow in fall.
You come at a great time (and a wonderful period) to explore Hungary. I am sure you will pick up materials both here and there to figure out how to spend 4 mos. Some suggestions (beyond Budapest): wine harvest in the Tokay area (Miskolc), peppers and paprika in picture-perfect Kalocsa, the Puszta (steppe) around Hortobagy, Balaton Lake (go in late Sept.) with wine areas (Tihany) ferry to Siofok. Castles: Fertöd, Esterhazy, Gödöllö. Eger mosque. Also Buda Hills and oak forests east (fall color).
Don't miss any folkdance performances! Weddings are often public w. live music. Excellent food, friendly service - what's not to love?