Latin first!!!
I find it very interesting to know about ancient Roman minds. Think of Virgil and Aeneid, of Cicero and De Amicitia, and of the excitement when you read those works and feel the beauty in each verse, each line, each word, and especially, the beauty of the language's preciseness. Think of your knowledge of humanity (history included), it'll grow as you gorge the flow of wisdoms from those minds.
French is a grandchild of Latin, so it becomes much easier if you are strong at Latin. And not only French, but all other Roman languages such as Spanish and Italian, will be. When you learn these languages, it is fun to compare how they are similar to each other, and how they've been derived from Latin.
Starting with French is understandably more fun, practical, and, possibly, more manageable. You can easily get to converse with French native and that is a huge advantage over learning Latin. But a strong foundation in Latin will get you much further with French, and any language, not just Roman ones, that you later decide to learn, as then you've had a strong understanding of language's general structure.
If you choose to take Latin, Vatican is a wonderful place to visit, because it is the only place on earth now where Latin is the official language, and it hosts a plethora of historical stuff, in Latin, and reflecting Roman culture.
You can visit these sites for more reference:
http://ancienthistory.about.com/cs/whystudyclassics/a/whystudylatin.htm
http://www.fll.vt.edu/French/whyfrench.html
Bona fortuna!