I have read that it's possible to 'hitch' a ride with sailboats and freight ships. But I haven't been able to find any sites that offer the sailing option.
Where is the best place to arrange this?
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There are several websites that you can use to find sailboat owners who want crew to accompany them on voyages. The largest is Find a Crew, and Crewseekers is an alternative. (I am not affiliated with either of these sites and can't vouch for the experience.) You can search for the routes, times and departure points you are interested in and then contact the sailboat owner making a trip that suits you. Or you can put your profile on the site and let sailboat owners contact you.
The basis of the arrangement is very much "hitch a ride". It's going to be up to the owner whether they want to take you or not, and you are usually expected to help out with the work (since that's why they want someone to ride along), but it's not going to be arduous unless you are making a challenging trip, and you can ask the owner what you are expected to do. You usually don't pay, except to share in the expenses. You should also be aware that sailboat itineraries are only ever approximate, and you can find that the trips is delayed or cancelled. Having some sailing experience helps but isn't always necessary.
Looks for yacht clubs, or marine schools (better if you are not near the sea).
People are required to do many miles on boats (sailing boats are common) in order to get high sea licenses. So they sell space on such boats, in order to share some costs. Usually it get from few days to one week, visiting many ports.