Contents insurance is highly recommended here in Cayman due to the risk of flooding and hurricane damage. Unless you live in a condo where strata insurance usually covers major kitchen appliances, these items should be included in your contents insurance coverage as they will not be covered in building insurance alone.
Be aware of exclusions to your coverage. Flood, earthquake or wind damage are usually not covered as standard items, these types of coverage must be purchased separately. Ask about any dollar limitation on claims as some insurers may limit how much they will pay.
Liability insurance is generally included with your policy, but it pays to know any upper limits on this amount of coverage. Ensure it is sufficient if you have significant assets.
Your Cayman Islands Real Estate agent is knowledgeable and will answer any of your questions regarding homeowner's insurance.