If the stars align as such that you stay long-term, buying property in the Cayman Islands can be a good investment. In the short-term though, pretty much everyone rents a fully furnished apartment/condo (usually on Seven Mile Beach).
The majority of rental properties are two-beds (but there is some flexibility if you want your own space or have a large family. Expect to pay the first month’s rent in advance and a damage deposit equivalent to 1 month rent.
Most of the rental properties are built and maintained to a high specification and it is the norm that residential complexes offer amenities such as pools and tennis courts. Prices will vary according to location and proximity to the beach but as a guide, 2 bed apartments start at around $1750pcm before bills and utilities (or $2700pcm if right on the beach itself).
As you would expect from an affluent jurisdiction such as this, if you have expensive tastes and want absolute luxury then it is achievable at the right price.