How did immigrants travel to get to Puerto Rico in the period from 1800-1870?
Where can I find information about the ships?
When I am trying to learn more about a topic or place and trying to find records that might mention my family or my research subjects, I follow a checklist I made for myself:
When people left France and came to Puerto Rico, there could be records in France, in Puerto Rico, or at other places the ships went to along the way, so following the routes the ships took is a good way to start looking (and interesting, too).
The Wikipedia article French immigration to Puerto Rico tells us some of the historical context. One important waypoint in the time line is the the Royal Decree of Graces of 1815. In its article on French Immigration to Puerto Rico Wikipedia says:
The Spanish Crown decided that one of the ways to end the pro-independence movement in Puerto Rico was to allow Europeans who were not of Spanish origin and who swore loyalty to the Spanish Crown to settle in the island.
Passenger lists often exist because someone passed a law requiring the ships' masters to record that information. What form the lists take depends on what the authorities wanted to know. If France wanted a record of people leaving France, records might exist there; if Spain wanted records of people moving to Puerto Rico, records might exist in Spain, or you could look for records in Puerto Rico. The records which the US Federal Government creates which are of historical interest are kept at the US National Archives (NARA); the records having to do with Puerto Rico are in the NYC Branch. Their Guide to Puerto Rican Records lists one group that may be of interest, Records of the Spanish Governors of Puerto Rico (Record Group 186) 1767-1880. One of the collections listed on that page is:
Microfilm copy of records concerning foreigners in Puerto Rico (Extranjeros in Puerto Rico), 1872-80 (19 rolls).
There is also a collection NARA publication T1170: Extranjeros (Foreigners) in Puerto Rico, 1815-1845, which you can browse online at FamilySearch: Puerto Rico, Records of Foreign Residents, 1815-1845.
NARA's description page also says that there was a fire in 1926 that gutted the Historical Archives in San Juan, so many of the records that were not kept by NARA were lost.
Finding Ships Records
Even if no one kept lists which tell us the passengers by name, or there were lists that have been destroyed, there are several ways to get clues about what transport was available at the time. Here are some resources:
Historical newspapers are a particularly rich source of information for shipping data.
Wikipedia's article on NewspaperCat: Catalog of Digital Historical Newspapers
FamilySearch Research Wiki: Digital Historical Newspapers
The 5 Best Free Sites for Online Newspaper Research for Genealogy posted on the blog The Ancestor Hunt, 24 Aug 2013 (which lists more than 5 sites)
Major subscription websites for newspaper research include:
Genealogy Bank (for tips on how to search for ships by name, see their blog post My Ancestor’s Trip to America: Newspapers Tell the Story
Newspapers.com (run by Ancestry.com -- a good deal of the content is licensed from Newspaper Archive)
For any information you find about the ships from these two lists, use that as a basis for a new search -- you can search for the ship's name, the shipping company, the other port listed in a ship manifest, and so on. Use any clue that turns up as a hint to look for more information. You can see how others have used these ship lists in their research by searching for the other questions here on G&FH.SE with the tag ship.