When we first arrived in Mykonos, we saw a row of windmills overlooking the harbour. It's one of the most popular and recognisable landmarks in Mykonos.
To get there, just follow the fishing harbour (the one in the town) around until you can see the wide, paved road leading up 200m or so to the windmills and the summit.
The windmills are pretty battered and work infrequently (check with the local tourist office on the docks), but the walk is gentle and you can get the whole top to yourself (in mid August we had the place to ourselves). There are great views down over Mykonos town and the surrounding harobour which is lit at night by the numerous bar and restaurant lights.
We found this to be a great picnic spot and a little shop about half way up the road sold cheap beer and alcho pops. This was a great find as most of the bars we checked out were too noisy and busy for a romantic rendezvous.