Where to go with 14 yo DD?

[Praminthehall]

Next summer I'd like to go away for around 10 days with my dd who will be 14. I'm finding it really hard to think of places which we will both enjoy and has a good mix of relaxing and things to do. We have been to the US in the past and she would be keen to go again but I would like to see more of the world and am up for a bit of an adventure ... although not an exhausting one as my job is full on and I would like to have a break.
Any ideas? I'm open to all suggestions.

What would / wouldn't you consider?

Do you want to be in one place or would you consider backpacking / travelling?

Do you want to go an see some 'famous' places ? - Maybe Italy ?
Or do you want to lie on beaches?
Or visit a big City... (Paris?)
Or both (Barcelona?)

Do you want to camp or go All Inclusive in a hotel or something in between?
Do you like heat?
Do you want to do something like windsurfing in the South of France, or walking in the Swiss Alps?
Or more cultural / historic - Normandy? Rome ?

Do you want to fly ? Or are you driving ? Or the train ?

Budget ?

Thankyou BackforGood for helping me to hone my thoughts!

I would consider a bit of moving around.

Famous places - yes
Odd bit of beach would be nice but we're neither of us after a wall to wall beach holiday
Some city good but we live in London so would prefer something different for the mostpart
Not camping because dd does a lot and would like something different for family holiday. We did all inclusive last summer and I felt like a fish out of water! And food not great

Not really looking for an 'activity' holiday - I don't think ...although cycling from place to place in an interesting area and at lazy-ish pace could be good now I think about it (if I didn't have to organise it myself!!)
Bits of history and culture good but need to be balanced out with less worthy stuff

Flying fine, other methods also OK. Prefer not to drive abroad although I think I am probably a bit of a wuss about that

Budget - flexible on this for right holiday. DD is my youngest and hasn't had an easy time recently, would like to invest in a special time together.

If US, places my 14y dd has enjoyed over there, other than Florida:

New York
Boston and Cape Cod
California - we flew to Vegas and then drove to Santa Monica; stayed by the beach but went into Hollywood/Beverely Hills, etc time at Disney and Universal, a trip to Warner Bros Studio Tour (which she loved as loves films/TV) as well as time by the beach, cycling, the pier, Venice Beach, etc.

She fancies Canada as not been before...or maybe China or Japan or somewhere similar.

How about Costa Rica. Fantastic beaches but also beautiful rainforests for amazing wildlife and a sloth sanctuary !!

Have you considered a cruise ...my 14 year old daughter loves cruising ...although I dont see much of her because she tends to go into the teen club ....(the teens tend to make friends very quickly)....We have travelled around a bit one summer but I wouldn't recommend doing this just for 10 days ...we spent a month doing Italy/Croatia ....

I was also going to suggest a cruise. We did one and although I didn't think it was worth the money the dc loved it and I know would go again. Plenty to do on the ship plus things to see at ports.

If you fancy the cycling there are companies that organise everything for you - we have used Cycling for softies before but there are others.

How about Lake Garda? Nice boat trips, good food and possible day trips to Verona and other cities. There's also a theme park nearby (Gardaland) and cycle routes round parts of the lake.