Honeymoon ideas to balance relaxation + exploration

Hi everyone,

My fiance and I are getting married in early September and are trying to plan a honeymoon sometime around late September to early October. She's very much into the idea of going to a resort, relaxing and sitting around a beach, but to be honest I find those kinds of vacations, especially resorts, pretty boring. I'd much rather spend time exploring, checking out historical sites and cities, etc.

We're looking for overseas destinations and trip ideas that would make for a good compromise. So far, we've narrowed things down to Italy (maybe a trip to Rome and the Amalfi Coast) or Greece (Athens, then Santorini, Mykonos and the like). We also have a Costco membership so we've been considering doing our planning through a Costco travel package to save ourselves some of the trouble of developing our own itinerary. Also, it would be our first time in Europe, and we're trying to keep our budget to around $5000 USD.

Does anybody have any experience with Costco travel, going to either of these countries, or things to do that would satisfy both of us? If there are other locations we should consider feel free to suggest them as well!