Our recent trip to Italy was not our first, so we didn’t feel pressured to visit all of the typical crowded (wonderful) tourist attractions in Rome, since we had already been there, done that. So our hotel in Rome was near the Museum of Modern Art. Who goes to a MODERN art museum in Rome?? LOL! It was a great museum, and not crowded. A very diverse selection of interesting art. I was amazed that we could stick our nose right up to a Van Gogh with nobody around and no ropes or glass in front of the painting.
If you promise not to tell anyone else, I will mention the area around Flagstaff Lake, Maine. 
Sometimes we go all weekend without seeing a single person. It’s a beautiful area. There IS a trail system operated by Maine Huts & Trails that sees more activity, but it’s pretty uncrowded, too. The Appalachian Trail goes through here, too. Some days we see no one at all on it. Occasionally, we get to talk to through hikers, which is cool.
For artwork, the Barnes Foundation in Philly. Incredible collection and they limit ticket sales to prevent it from getting too crowded.
You can stick your nose in Van Goghs, Matisses, Picassos, etc. plus the world’s largest collections of Renoir (181) and Cezanne (69).
It’s a gem.
Any BLM land camping is less crowded, open April to October in the mountains.
Christmas Day in NYC. Helsinki in March.
In France: le clos lucé near Amboise in the Loire Valley - Leonardo da Vinci’s residence, lots of da Vinci works (inventions, drawings…) to see.
Going south across the Loire valley, the Loire valley wineries for white wines, then west to La rochelle (site of the siege in the three musketeers, a real major battle in the French civil war between Catholics and Protestants, la rochelle being a Protestant stronghold.) Late July to late September is okay (early July you have festivals), totally fine in May and June. Lots to see.
Drive South along the coast (or take the bus) for about one hour. Stop in Royan. It’s the site of the first “real life” test of napalm by the US army trying to root out Nazi who wouldn’t leave, thus levelling select parts of the town which now boasts classic 1920s villas and post WW2 urban planning - probably the only successful one because it thought of people first. Also the only brutalist construction that I find bearable, the cathedral. Architecture tours are worth it. It’s a resort town so crowded July 14-Aug 15, but even then reasonable compared to what you’d see on some beaches along the Eastern seaboard. (I’m talking ‘circle the town square three times to find a parking spot or park with no fuss at the station and walk for 10mn.’)
Take the ferry then drive down to Bordeaux. It’s a major city so it won’t be empty but will be fine in August when locals are in Arcachon. Dont miss Place des Quinconces and the wine museum. Then St Émilion, likely crowded but worth it for one afternoon (good shoes required, wine tasting optional).
To the Mediterranean: cross the Larzac (which the French call a ‘desert’, but really is a moor). The Millau Viaduc is worth seeing. Try Roquefort cheese if you dare. Also less crowded : Sète and l’etang de thau (brackish water where a significant percentage of the French mussels and oysters come from), Béziers and its cathedral, Montpellier, Aigues Mortes (another Protestant stronghold where locals are proud of a woman who refused to convert and spent years jailed, scratching into a stone with her nails the word ‘resist’.)
Those are worth visiting and are rarely on Americans’ maps.
Castle Valley, Utah.
Ohhh… good one. Wine lovers… if you despise the crowds and high prices of Napa/Sonoma, come to Walla Walla or Richland Tri-cities area! We have lots of wine!!!
We recently spent a lovely weekend in that area. Stayed at the Lodge in Richland. Water view, top floor room was $200 plus tax. Truly to find that kind of weekend pricing in Napa. Ha! A couple of nice fish restaurants right next door. There is a river dinner cruise boat (looked very nice but it was sold out when we were there) that departs from the pier right under the hotel’s widows. Lots of wineries… lots. No French Laundry type of restaurants though. Some day.
Speaking of vineyards, there are are a few of them along Oak Creek between Sedona and Jerome AZ. They are all on N. Page Springs Road. Only Page Springs Cellar is fairly “large”, and serves good food too, but I’ve been there a few times and never found it overwhelmingly crowded. No tour buses around, and the weather is fine all year.