Games: Any setting where strategic agents interact to produce a collective outcome. E.g. tic-tac-toe, rock-paper-scissors, buying/selling objects, applying for jobs/hiring decisions, ride-hailing apps and much more.
Algorithmic Game Theory (AGT): The theoretical study of games and the algorithmic aspects of finding the outcomes of such games under strategic behaviour.
Computational Social Choice (CompSoC): Sub-field of AGT focused on settings without money where agents submit preferences and a collective decision has to be taken by a central planner/preference aggregator. E.g. voting/elections, choosing where to place bus stops/factories, dorm assignments, college admissions, refugee resettlement and much more.
If you're interested in pursuing research on computational social choice and/or algorithmic game theory and would like to gain some background, it cannot hurt to do a course introducing you to this topic. Some of online options (with video lectures) include:
Felix Brandt's course on Computational Social Choice
Tim Roughgarden's courses on Algorithmic Game Theory and Advanced Mechanism Design.
Crash course on Market Design and Computational Fair Division at ADFOCS2020.
Rohit Vaish's course on Computational Social Choice.
When doing research in these areas, you will benefit from some technical prerequisites that are widely offered at universities such as:
Algorithm Design courses: this ranges from basic courses on data structures and algorithms to those on approximation algorithms, parameterized algorithms and randomized algorithms. I would consider a basic algorithms course as necessary to start research and the rest as topics that can be pursued simultaneously to your research.
Toolbox courses: While some universities have recently started offering courses called theorist's toolkit/toolbox, broader courses on probability theory, combinatorial optimization, concentration inequalities can also be useful.
In general, any course helping you get acquainted with writing formal proofs can be helpful. This includes courses on Analysis, Complexity Theory, Optimization etc. As such, these courses are not critical, but can be instrumental in helping improve your foundation.
Finally, while research can certainly be pursued independently, it helps immensely to have someone guide you into the field. This can be at an undergrad, masters or PhD level. For undergrad and masters students, if you don't have someone in your university who work in this area, you can reach out to folks working in this area. They will usually have time to collaborate and even sometimes some funds to host you at their home institution. I would even recommend that PhD students whose advisors already work in this space to try and collaborate with other folks, as it is very useful to learn from different work and writing styles.
While there are very many researchers in this field, I can happily recommend some of my collaborators as kind and helpful folks to introduce you to the field. This includes: Rohit Vaish (IIT Delhi), Sanjukta Roy (ISI Kolkata), Hadi Hosseini (Penn State), Ioannis Caragiannis (Aarhus University), Siddharth Barman (IISc), Umang Bhaskar (TIFR), Ganesh Ghalme (IIT Hyderabad), Shweta Jain (IIT Ropar) and Arpita Biswas (Rutgers University). This list is no particular order and people left out of this list are not necessarily bad. I just have not worked with them, so can't comment on their advising or are a collaborator who's not yet a faculty.
Disclaimer: Different people have different working styles, so because I enjoyed working with someone does not mean everyone on the planet will or that they will necessarily be a good collaborator for you.