A day trip that would satisfy the cultural component of your Mykonos holiday would be a visit to nearby Delos island, to view the historic ruins and see the archaeological museum. Boats to Delos leave from a quay in the harbour at Mykonos Town at three different times in the morning, returning between noon and 3 pm (leaving you plenty of time to explore Mykonos Town).
There is a huge array of dining options on Mykonos. See this thread for some information about what's there: tripadvisor.com/ShowTopic-g189430-i401-k5465…
In Mykonos Town, restaurants you might enjoy include Avra, Bakalo, Catari, Interni, Roca Cookery, To Ma'Ereio, Fato a mano, and Uno Con Carne, to name just a few. Right at Agios Ioannis, a short walk from the hotel, check out Hippie Fish and Pyli .
Can't advise about the hotel shuttles, but can tell you there is regular bus service between Mykonos Town and Agios Ioannis. At the time of season you're travelling, taxis shouldn't be too difficult to obtain. The bus and taxi rides to/from Town will be short -- 10to 15 mins. for the bus and less for the cab.
Your hotel is reasonable walking distance from Ornos beach, where you'll find a good selection of restaurants and bars along the beach.
I'm sure the staff at the Grand will be able to give you good recommendations on things to see and do ... they are renowned for their excellent customer service.