When I'm beginning research in an area which is unfamiliar to me, I generally start with the Research Wiki at FamilySearch.org.
The main article is Cuba Genealogy -- for research strategies, see the Cuba Quick Start Guide, which says:
Unfortunately for those with ancestors from Cuba, FamilySearch does
not have any microfilmed or digitized records for Cuba.
Their pages on Cuba are not very extensive, but there is an article Cuba websites, including CubaGenWeb.org. To see other articles such as Cuban Family History Resources search for the wiki category Cuba. Remember, the Research Wiki started out with contributions from FamilySearch and is added to by volunteers, so it pays to go through every article to glean every bit of information you can find.
Just like when you are searching for a person, if searching for the country fails, try broadening the search to see what is available for the region.
Another good place to look for resources are the big directory sites like Cyndi's List: Caribbean / The West Indies » General Resources
Some websites which may be of interest:
Where there is no big "one-stop shopping" site to get you started, it helps to get creative with searches. Look for records, yes - but also look for research guides and finding aids, like the Overview of Hispanic Research published in Ancestry's Research Wiki. Read genealogical periodicals and blogs, and see what sources other researchers used while doing their research. Don't just look for things which might mention your family members -- look for anyone researching in the same time and geographical area, and read their work carefully, giving special attention to the bibliographies in published genealogical periodicals.
Doing general research on a locality can also yield clues. Don't limit yourself to genealogical sites and genealogy societies -- information from a historical society's website or periodical can help you correctly interpret the details in any historical record you can find.
Online Catalogs that might help:
Keep an open mind while searching, because just like people, historical records and material about a country can end up far away from where it was created. The US National Archives has Records of the Provisional Government of Cuba. Will there be material here which is of genealogical interest? Maybe not -- but if you had someone who died of yellow fever during the period of the US Provisional Government, the statistical reports might be of interest to you.
Lisa Louise Cooke's presentations on using Google for Genealogy are quite good:
I found these results via searching Google in the USA, in English. Spanish speakers may be able to find better information by switching to searching to Google Cuba and searching there.
Making a research plan with specific research questions is a great way to focus on what information you want to find and where you might be able to find it. For other ideas, see How can I determine what records are available in a particular locale?