I heard someone say "if (as a speaker of English) you learn French, you get Spanish for free".
Is this true? Is Spanish mostly a subset of French?
That's a very dubious assertion; I know for certain that speaking French fluently is not going to make a conversation in Spanish intelligible at all to you, nor one either in Portugese or Italian. It must be taken very generally: you'll find often the same latin roots but that's about all in the way of a solid similarity. For instance, whereas in French and English you have just one verb "to be" in Spanish you have two (ser, estar); you find the forms of the French verb "être" (sont, sera, serais, être, est, …) in those two verbs but very much modified and that's not going to help: the precise combinations of those forms are made according to quantities of different rules; for instance when you use personal pronouns with verbs in French, you don't use any in Spanish (« Il parle fort. », « Habla fuerte. »). Here you do find that "fort" is "fuerte" but in spite of the same latin root, you'll never know that before you've actually learned this correspondence.