I'm thinking about doing a master's and considering subjects; it seemed great to do something related to the Vikings. I already speak French but I'd like to know what else I need to learn in the next year and a half to get ready. If you can point out "gaps in the literature" that might provide leads for research projects, I'd really appreciate it too.
If you are examining Old Norse texts, it is incredibly useful to understand some nordic languages. It's hard to actually learn Old Norse nowadays, as there are few comprehensive resources, but if you're really set on it I'd look https://notendur.hi.is/~haukurth/norse/ and https://archive.org/details/EnglishOldNorseDictionary .
If this is too taxing, which it was for me, I'd suggest that it's really worth learning Icelandic, Swedish or Norwegian. (Maybe Finnish, but I've never attempted to learn that so I don't know). The latter two are very similar to each other. Icelandic words are very similar to their old norse counterparts when written, but the pronunciation has changed dramatically. Something like Duolingo is good for starting to learn a language, but as far as I know they don't offer icelandic, only swedish and norwegian.
As an aside, when I started learning norwegian I noticed a lot of similarities with my family's dialect of english, which has a lot of norse influence. The whole thing helps you gain a better understanding of germanic languages, and you may find you're able to decode parts of languages you haven't even learnt.
Best of luck with your master's - the ancient scandinavians are fascinating to study, and I'm sure you won't run out of material!
It really depends what you're going for. There's lots of great options. With French beneath your belt, one obvious choice is Latin, which will let you dig a bit deeper into the Carolingian chronicles of viking activity. Old Norse is a good option if you want to research the sagas or Norse mythology. These are exciting areas of study and Old Norse can be a lot of fun, but keep in mind that they often tell us much more about when they were written down (generally post-1200) than about the Viking Age (ca. 800-1100). Arabic is another interesting option, given the interest of Arabic-speaking travelers, geographers, and historians in Norse activity in both Iberia and eastern Europe.
For modern scholarly languages, any of the Scandinavian languages would be a good choice and help deepen your research, although I don't think they're immediately necessary. Scandinavian scholars tend to publish lots and lots of articles, so it's generally possible to find at least summaries of key evidence and arguments in English sources. German is another good option. German scholars are less prone to publishing in English, and some key Scandinavian academic work is published only in German. German is also a good gateway to other languages, and getting to know a language that relies heavily on noun cases would be a good complement to your knowledge of verbal tenses from French. Russian is a much-needed language among western scholars of the Viking Age. Since the end of the Cold War, there's been increasing access to Russian archaeology of eastern Norse activity, but Russian is no longer emphasized in western schools, and so this information never really gets disseminated. Polish and Finnish are also worth considering. These will narrow the focus of your studies, but that can actually be a good thing for masters work, and Poland and Finland might be more attractive than Russia for some American travelers at present.
I'd recommend really thinking about why you want to learn this language. Learning a primary source language is what makes doing history so much fun, in my opinion, since it gets you more into the nitty-gritty of the past. It's also something that very few people do, so it's actually a fairly unique experience. But learning a modern language might open doors for careers that are only tangential to an MA in history. For the same reason, I'd recommend thinking about integrating digital analysis into your masters work. Try googling Digital Humanities or ask a meta question here about what digital projects are going on at present.
Language Learning Resources
Jesse Byock, Viking Language, 2 vols. This is an excellent introduction to Old Norse as well as general Viking Age history and archaeology, plus it's a lot of fun. Highly recommended.
Wheelock's Latin. This is my favorite Latin primer, and it's possible albeit difficult to work through on one's own. I'd generally recommend taking a class instead.
Sandberg's German for Reading. Sandberg also wrote a French volume. Both are a bit tough to find in print, although pdfs circulate in dark corners of the internet. This skips the conversational stuff (including pronunciation, sadly) and goes instead straight into academic reading, which can be acquired with these texts in just a few months.
Duolingo. Don't discount apps like Duolingo. Although they won't let you plunge into academic texts, they're great prep for taking a course or visiting a country for research.
Research Ideas may be found in some of the texts below. Pick and choose as you will. Primary sources can generally be found free online or for cheap from a book seller. Secondary sources might best be found through your local or university library.
Brink and Price, The Viking World (2008). An absolute must. Still a generally current introduction to all major aspects of Viking Age scholarship currently under way.
Andrén et al., eds., Old Norse Religion in Long-Term Perspectives (1996). A really interesting volume full of short articles. Some are still brilliant, others are aging, and a few should be read with a healthy dose of skepticism.
Price, The Viking Way (2002). Although not everyone agrees with Price's conclusions, this work has irrevocably shaped the field for how we study religion during the Viking Age.
Svanberg, Death Rituals in South-East Scandinavia (2003). A hearty introduction to cemetery archaeology, which I think shows some of the diversity of religious perspectives that undermine efforts to reconstruct a "Norse religion" from bits and pieces that probably never actually went together.
Franklin and Shepard, The Emergence of Rus (1994). The story of Norse activity in the East.
Penguin's Ibn Fadlan and the Land of Darkness (2012). A great collection of primary sources on Norse activity in the East.
Barrett, ed., Contact, Continuity, and Collapse (2003). The story of Norse activity in the Atlantic West.
Christys, Vikings in the South (2015). *The corpus of sources for Norse activity in Iberia.
The Anglo-Saxon Chronicle (any edition). Originally written at King Alfred's court in the 890s, this was meant to be a document of why England should rally to him rather than becoming Norse, and copies of it were regularly updated for centuries, making this a rather hodgepodge of texts and completely invaluable for study.
Scholz, Carolingian Chronicles (1970). My preferred edition of two key sources for Francia during the early Viking Age, although vikings don't play too large a role.
Dutton, Carolingian Civilization: A Reader (2004). A great compendium of primary sources for France for the whole of the Viking Age, and it includes selections from Scholz.