2 Week European Vacation

You might want to browse the "Explore Europe" link in this website to see what appeals to you, keeping in mind that the cities it features tend to be popular with at least RS fans and probably many others too. Don't expect much in the way of "hidden gems" from any website or guidebook though -- if they're listed in such sources they're not exactly "hidden."

Some places I've especially liked, not "hidden" or "unfrequented" but also not in the "blockbuster" category, would include (in no particular order) Durham, Chester, Wells, York, Nafplion, Delphi, Padua, Cahors, Bayeux, Chinon, Beynac, Beaune, Bratislava, Haarlem, Aarhus, and Tallinn. Chania in Crete is supposed to be wonderful (my wife agrees) but I haven't been there.

An itinerary built around all the above would take much longer than two weeks and wouldn't make much geographic sense. My last thought would be that, with only two weeks, you might limit yourselves to two or three big cities, or maybe four small cities, in one or two countries. Moving around costs time and money, and your time at least is short.

I would actually suggest picking an area and then digging into smaller cities in that region. Instead of trying to hop from France to Italy to Bulgaria in search of a series of "unfrequented cities", dig into one area. I can suggest some that I am familiar with, but there are so many places!

-Greece's Peloponnese or other parts of the mainland. Athens and islands like Santorini are, of course very famous for good reason - but there's so much on the mainland outside of Athens.

-Sicily or far south Italy. In Sicily, Taormina and the beaches get pretty touristy. The rest of the island gets its share of visitors, but nowhere near the amount as somewhere like Rome. The Ortygia island of Siracusa, baroque towns of Noto and others nearby are great. Greek ruins in Agrigento. Palermo for amazing food and architecture (fascinating mix of Italian, Norman, Arabic, and others).
Then there's the far south of the mainland Italy. I haven't been but Lecce, Matera, and Puglia are all on my list.

-My favorite region isn't exactly unknown - Andalucia, Spain. The beaches are extremely touristed (most not worth visiting in my opinion). Cities like Seville, Granada, and Cordoba are wonderful - but they aren't unknown. There are smaller towns like Ubeda or Jaen that are more off-the-radar.

-I have zero knowledge of Eastern Europe outside the cities like Prague and Budapest. My sense is that if you explore smaller cities in the East you will run into less tourists and have plenty of discoveries. Bulgaria, for example, is somewhere I am interested in.

Thanks for your input.
One or two countries looks best to focus on for time and cost, given our $6K budget. A round trip is preferred if practical versus cost of an open jaw ticket.
I've been to several cities in England and Italy. Plus Amsterdam, Brussels, Paris, Barcelona, Valencia, Sevilla, Lisbon, Porto, Frankfurt, Vienna, and Prague.

Getting around depends on efficiency of time and logistics versus cost. Probably a combination of car, train, and air.

Why not try Poland. You can easily spend two weeks there with a combo of small and large cities. The train system is easy to navigate and will get you wherever you need to go. It is so very afforable and there is plenty of history, music (Chopin), great food, and nice people. We spent two weeks there last July and travelled from Krakow to wroclaw to Gdansk (day trip to Marlbork) to Warsaw. It wasnt as crowded as other European cities, such as Prague, and is a hidden gem in my opinion.

You have mentioned a couple of times the cost of an open jaw ticket.

That's surprising - it should be close.

Are you looking at the Multi-city feature of the airlines or pricing your open jaw as two one-way tickets? If you do that, yes you will likely suffer. If you use the multi-city feature, or whatever your airline calls it, there should be little if any excess costs.

Unlike many parts of Europe June isn’t peak season in France. Two weeks would allow you to fly into Paris and head out to Normandy/Brittany, then visit La Rochelle, which has great food. From there, you could head to the chic Ile de Re or the Loire valley for wine and chateaux before winding your way back to Paris.

I'm guessing Travel Man means that USD 6000 is equivalent to about EUR 5400 -- NOT that changing money will actually cost you $600 in fees. You need to start thinking in € when making plans and reservations.

I spent about 10 days in Hungary last September, in Budapest and lovely smaller cities - Pecs, Kecskemét, Szeged. Budapest checks all your "wants" and the smaller cities had very interesting architecture. None had many tourists and Hungary's a fairly cheap destination with great food and wine as well.

Following on the excellent suggestions of Chani, Barbara, and Jennifer: the easiest way to accomplish what you are looking for is to pick one country (or two countries in a small region), start with the famous capital and other "brand name" cities and areas, but then go outside them.

So:
Go to Poland and see places outside Warsaw, Krakow, and Gdansk.
Go to the Czech Republic and see places outside of Prague and Cesky Krumlov.
Go to Hungary and see places outside Budapest.
Go to France and see places outside Paris and the Riviera.
Go to the UK and see places outside of London, York, and Edinburgh.
Etc.

Note that the specific places I mentioned are all worthwhile, but getting out of them will give you the "unfrequented cities" experience you desire. For instance, I believe something like 90% of all tourists to the Czech Republic never see any place in the country but Prague. I really liked Prague, but I also enjoyed Olomouc (covered in Rick's book). In Hungary, I enjoyed Budapest, but also enjoyed Pecs (in Rick's book) and Vac (not in Rick's book - I found it in Frommers).

If you want a more specific itinerary, you'll be very interested in my trip report about Glasgow, Manchester, and Liverpool. The first two get few tourists, and most tourists to Liverpool seemed to be from other parts of the UK, but I had a great time in all three: https://community.ricksteves.com/travel-forum/trip-reports/uk-trip-report-glasgow-manchester-liverpool-in-september-2016

If you tell us where you're flying from, we can give tips on flight options. But Nigel has already given the most important one: always look for "multi city" or "multiple destinations" when flying open jaw, rather than two one way tickets. There are exceptions, but many airlines charge much more for two one-ways.

There are plenty of areas in every country which are along your lines.

We were in Bretagne, France (Finisterre) in 2017. Very nice, not huge sights, but pretty sea-side villages, WWII beach fortifications, walled towns, etc.

Germany outside of Berlin in the East - Leipzig, Dresden, Cottbus.

Croatia/Slovenia/Serbia/Bosnia-Herzogovenia.

I just saw Harold's comment. I too was in Liverpool/Manchester area in 2017. Very pleasant. Lots to do - Chester (Roman town), Liverpool (Beatles), Manchester (all about the futbol), etc.

I expect it all depends on your home (or nearby) airport, but I know I rarely spend more than $100(cdn) for an open jaw ticket vs return. Even for 2 people, that barely makes the hassle of heading back to somewhere convenient. Your time is also valuable.
You mention you have been in areas of Italy, but have you spent time in Emilia Romagna or the Veneto? There is a lot of secondary and smaller towns with lots of exquisite art, architecture, and cuisine in that region. Parma (as in ham), Parmasan cheese, Lambourgini and Ferarri car factories, Palladian mansions, Giotto frescos just to name a few. Granted, this location is well known to tourists, but is so rich in things to see and do, and not on the 'first timer' radar, that it might be worth it. Bologna is often the hub city for day trips, but head over one stop to Ferarra and it will be much less touristed but just as central.
Head further east to the coast and Ravenna has world class Byzantine mosaics. Some like San Marino (a tiny country). Though I haven't been yet, just south of there is a cruise port - Ancona - but it looks quite appealing with ruins, sunny walks. It seems to only be on the radar of European cruise lines so nowhere near as touristed as Venice or Rome. Looking at the port schedule, there are numerous days in June with no cruise ships so you can have the facilities without the cruisers.
Then head down the Italian spine and visit hill towns if you want to do a road trip.
Or, do the Italian area (fly into Milan, Bologna or Venice) and take one of the scenic trains into Switzerland to see the Alps and mountainous towns. Switzerland will eat more of your budget, but if only for 2 or 3 days, it could be lovely (for more of the train details, 'research' Switzerland sights or guide books for the few routes that head into Italy).