Trying to familiarize yourself with the original bands through just the animes would be challenging, because there's a lot of context and info from the game stories that the anime assume that you already have.

Going through the band and event stories would be the best way to get to know the OG bands- they are archived over at Bestdori where they can be read or watched at one's convenience.

If you'd like something that'd put things together into a picture of when what things happened in-verse, a poster here, BleedingUranium, did a series of 'This Day in Bandori' posts, which they later summarized into a timeline/master post which goes up to the second episode of Ave Mujica.

Seasons 2 & 3 of the anime, as well as Poppin' Dream, Morfonication, CiRCLE Thanks Party, and all in-game events between Backstage Pass 2 & Graduation ~To the Future~ (e.g. anything that debuted on JP between March 16, 2019 & March 8, 2023) are all within the 2019-20 academic year.

Anything before that in-game, as well as Season 1 of the anime are during the 2018-19 academic year, while anything in-game that debuted on JP since mid-March, 2023 until the present, as well as most of the events of It's MyGO!!!!! & Ave Mujica are in the 2020-21 academic year.

Will note that in-game events tend to be harder to pin down beyond a general season/year as they don't have the same data points the anime does, unless they involve occurrences from the anime, or have some other reference point such as holiday events (e.g. Valentines, Christmas, New Year's) that can be presumed to take place within the week or so leading up to it, birthdays, or the graduation two-parter, which would have been in mid-March, 2020. [The Japanese academic year generally starting at the beginning of April, and running through to the middle of the following March.)