Shipping a bike overseas has become super expensive in the past few months. I considered, for my recent month long trip to Germany, but decided to buy a bike in Germany instead.
It's staying there, for future use. An advantage I have.
I traveled extensively, all over the US, with a racing bike in bike case, for over 15 yrs. Was a PITA, and I was much, much younger. I can;t imagine how much more complicated it's become - and it has become more complicated.
Cars in Europe - most rental cars, except for some of the very largest, would have a difficult time fitting one bike with some dismantling, much less two or more, and some amount of luggage. So an external bike rack might be needed... Folding bike might be a bit easier. On this recent trip I had a rental Audi Q2, and no way would a normal road bike fit well inside, without some serious effort to minimize damage to the car interior.
If you're traveling US by vehicle/van/land barge then prolly ok if you can secure on roof or internal. Bike thieves have become real pros.
If you're considering the US west coast, Arizona/New Mexico or Utah (Utah having more winter conditions), do consider some extended stay time on the Cal. Central Coast to San Francisco area (and a bit north of SFO). Winter riding is great out here! Many pro teams use our area for Winter training, camps. A huge variety of riding types, warm daytime temps with abbreviated periods of rain.
Where I am, the Carpinteria/Santa Barbara/Goleta valley is a gateway to riding here and all the way up to San Fran. We also have a large number of ride groups and rides for many levels of riding.
We do get quite a significant number of snowbirds who spend extended time here.
Culturally, there's always quite a bit happening here; so cycling is not the ONLY thing.
Ride On
Yuri