The Mask of Arcadius you install is really a combination of the first two entries in the series, First Arrival and Mask of Arcadius; the latter was released as an expansion rather than a separate title. Just launch Mask, and upon selecting New Game choose First Arrival. After you progress far enough, the game will seamlessly transition to the Mask content; there's no reason to ever select Mask from the New Game menu unless you're replaying the game and just want to skip the first part of the narrative.

Mask of Arcadius ends on a cliffhanger, and Liberation Day picks up immediately where it leaves off. Upon completing Liberation Day, you'll unlock the branching [Re]turn DLC, a series of what-if? scenarios that are all non-canon yet still shed insight into the characters, scenarios, and lore of the series.

Academy takes place in an alternate universe from the main series and is largely unconnected to it beyond Easter Eggs and inside jokes, though there is one major exception that ties the two together in a number of ways.

Mask of Arcadius (made up of two parts (First Arrival (1) + Mask of Arcadius (2)) -> Sunrider Liberation Day (3).

Sunrider Academy is then a spin off so it doesn't matter where it is played although I recommend that you do so after MoA and before LD.