The lecture notes by Frank Nielsen are succinct and fairly self-contained and maybe good to get a first idea. The books [1,2] by Amari can serve for a more in-depth study and contain a fair bit of differential geometry background. In order to get a flavour for some of the applications, I would suggest .
EDIT: Some additional remarks on the differential geometry background
It is quite easy to get bogged down here. I would suggest first reading ref. 1 which has a nice overview of the most important concepts in chapter 1, and then reading additional texts as the need arises (see the comment by @Gabe below). In my view, it is important to have a good grasp of the fundamentals of smooth manifolds and have the geometric intuition for connections and metrics as additional structures on the manifold, i.e. what purpose they serve. It is good to have some experience in Riemannian geometry, enough to understand where and how dually flat connections are different from the Levi-Civita connection. Less important, at least for a first contact with information geometry, are all the global aspects, fibre bundles etc. Of course, if you are interested in those topics for their own sake, there is a lot of interesting stuff. Personally, I learned a lot from  but any book which covers the same range of topics would be fine, there are many texts which cater to different styles/preferences. For more guidance on geometry textbooks, there are a bunch of related questions here on MO.
- Amari, S., & Nagaoka, H. (2007). Methods of Information Geometry. American Mathematical Society.
- Amari, S. (2016). Information Geometry and Its Applications (Vol. 194). Tokyo: Springer Japan.
- Nielsen, Frank, Frank Critchley, and Christopher TJ Dodson. Computational Information Geometry. Springer: Berlin, Germany, 2017.
- Lee, J. M. (2009). Manifolds and Differential Geometry (Vol. 107). American Mathematical Society.
Please let me know if you need further or more specific suggestions.