The first question we had in planning our destination wedding was whether the marriage would be legally recognized in the United States.
The U.S. State Department actually has a website for U.S. citizens interested in getting married in a foreign country.
You need to follow the legal procedures of the country where you will be married. Most countries have residency requirements (the length of time before the wedding you need to be in the country). In the Bahamas, the residency requirement was one day. However, Tahiti has a residency requirement of 30 days, essentially eliminating it as a potential location for destination weddings.
You will need to produce legal documentation (usually your passports), but could also include birth certificates, death certificates (widow/widower) and/or divorce decrees. Since this was the first marriage for both us, we only needed our passports and copies of our birth certificates. In the Bahamas, we also needed to go through a brief interview with a local official (about 10 minutes). We did not need to do a blood test, which can be a major issue for some people getting married in some countries (including Mexico).
The other major issue that people encounter is getting their marriage certificates/licenses translated into English (which is required for your marriage to be legal in the U.S.). The official language of the Bahamas was English, so we did not have any troubles. We wisely obtained multiple notarized copies of our license, which came in handy when we got home. We needed extra copies of the license for the name change.
Destination weddings can introduce some additional hopes you might need to jump through with respect to legal issues, but it is not a major problem.