Sorry but your question is way too general. In August/September it will be warm pretty much anywhere North of the Tropic of Capricorn and south of the 54th parallel. Since that's a hugh area that encompasses about 2/3's of the world we need a little more info from you.

Where are you traveling from? it will make a big difference on where you can go on that budget. How far are you willing to travel? Do you want a land based itinerary or perhaps a cruise? Is there a particular country you want to visit and if so any specific area in that country?

How long are you willing to spend traveling from your point of origin (wherever that might be) to reach your destination? What modes of transportation (car, train, bus, plane, boat or combination of those) are you willing to take?

Where have you traveled to in the past? What did you like about those trips? What did you dislike?

Honestly, you can find "food" everywhere so you have to tell us what kind of "food" you are seeking - gourmet meals, local dishes, home cooking, comfort food???? And, you have to define what you mean by "non-touristy activities". What types of activities interest you?

Although your questions are too general, I would love to visit Italy in August. Foods are great along with Italian culture and history. And weather is warm.