You are transferring ONLY the autosomal DNA results to FTDNA. They do not transfer Y-DNA but you can post your results from other companies to Y-Search, their free public database. A few surname projects are beginning to accept autosomal results but not all.
Which test to take first? What do you want to find out?
Are you trying to expand your father line? Do you have several possible surname families but aren't sure which is yours? If so, take a Y-DNA test through a surname project and discuss your particular case with the administrator. Most are very knowledgeable and helpful and will suggest which test or tests to take first.
Do you already have a well documented - NOT copied from other people!!! - family tree for 6+ generations with some gaps? An autosomal test will potentially be of great use. I have had the best matches from my ancestors who arrived in America prior to about 1750. This is because Ancestry is only selling the test so far in the US so fewer recent immigrant families are included.
If you don't have a tree, do that first! The most useful Genealogy DNA test is one from the company who tested your distant cousins. Comparing results is the only real value to DNA testing. A DNA test is most useful in confirming the paper documents and conclusions you have already made.