Early June is still very much spring. Pick something where you don’t have to pack a thousand ‘maybe’ clothes, and maybe just need a lightweight wind jacket. "just in case’ stuff means over packing and carrying a lot of stuff with you.
I’d recommend some place it’s easy to find food/water as you’re riding. If you get too adventurous, you’re 30mi from a town and out of water. Also pick a place where you know you’ll be exploring and when you’re not exploring. Hiring a guide (guide or kid from a local bike shop) isn’t a bad idea even if the location is pretty approachable. Also, keep in mind that you’re only biking x hours /day, you’re chilling the rest. It’s better to dip your toe into adventuring with a group then it is to go full Laos.
Recommendations… remember you don’t travel to France, you don’t travel to Paris, you travel to a particular part of Paris - you can go to that spot in Paris and never see “France”. I never biked there, but northern Italy based out of Florence. Have someone drive you out (or train) to somewhere everyday and ride back to the hub. You can hit the Cinque Terra, Sienna/"Tuscany, a finger lakes day (keep the main room in Florence and an overnight room at the daytrip for showers), etc. That gives you ocean, hills, weather, flats, and some convenience - more variety is better than less variety. Maybe pick another and you’ve got a full trip.
Ignoring the cycling portion of the question, if its your first time in Europe try and visit Rome and Paris.
Really? Rome is one my all time travel let downs. Been there twice, checked out of my hotel two days early both times. You see the Vatican, the forum,… and pretty quickly you’re looking at yet another church and god statue. The eateries aren’t plentiful, the neighborhoods aren’t fun to explore…there are better places to be. For urban Western Europe - Berlin, Paris, London, Amsterdam, and Barcelona are legit. Rome is up there with Brussels, Frankfurt, and some airport corridors.
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Actually, that would be a GREAT trip… get a Europass, explore the city one day by bike, go for a big ride the next, train ride to the next town, and repeat.