There are three possibilities in spoken English: 1. the first of August 2. August the first 3. August first Both 1 and 2 are normal in British English. Number 3 is never used by British English speakers. I believe that 3 is standard in American English, while 1 is reserved for formal and written contexts.

Both are right, but "First of August" is more formal. To friends: "Do you have the day off on August 1st?" Written in a wedding invitation: "Please join us on the first of August to witness the wedding of John and Mary"