I think what is being missed here is the second game in the series, Danganronpa 2: Goodbye Despair.
From the Wikipedia article:
It is the second installment in the Danganronpa series, and a direct sequel to the 2010 game Danganronpa: Trigger Happy Havoc.
It's the sequel to Danganronpa 1, adding to the DGR universe beyond the events of Trigger Happy Havoc, complete with a whole new set of characters, scenery, and more.
Danganronpa: The Animation is an anime adaptation of DGR1, AKA Trigger Happy Havoc, AKA THH. THH is the basis of the entire DGR story/franchise; it is the first game, but not the first event of DGR chronologically. Essentially all other canon media in the DGR universe builds upon the events of THH, so you'll want to start here. Either the show or the game will suffice, though I recommend the game.
DGR2's story is critical to both sides of the DGR3 animes. As per the wiki article, Despair
tells the backstory of Hope's Peak Academy's 77th Class
which is the same class featured in DGR2. Even though 3/Despair happens chronologically before DGR2, I recommend playing DGR2 before watching 3/Despair. DGR2 tells the story of the 77th class, which is then given more context/backstory in 3/Despair. DGR2 does not have an anime adaptation, but it apparently has a manga.
The Future and Despair arcs aired simultaneously in an alternating fashion, starting with Future. Chronologically, Despair happens entirely before Future, but the intended watching order is actually the order in which the episodes are released; so, also in an alternating fashion starting with Future (source: wiki.) In other words, Future ep. 1, then Despair ep. 1, then Future ep. 2, then Despair ep. 2, so on and so forth.
So, to hopefully answer your questions:
All Series Watch Order:
- DGR1/THH/Danganronpa: The Animation
- DGR2: Goodbye Despair (game or manga)
- DGR Ultra Despair Girls (game)
A few characters from UDG appear in Future, so it may benefit you to play this first, but it's not a dealbreaker like not playing DGR2 is.
- Alternate watching DGR3/Future and DGR3/Despair episodes, i.e. F1 -> D1 -> F2 -> D2... etc.
- Danganronpa V3: Killing Harmony (game)
IMO, V3's only strict prerequisite is DGR1 to get the full picture, but I recommend experiencing the whole series to be safe, if you can.
The DGR subreddit also has a good FAQ/writeup on this topic, including Supplementary Material.
- DGR3 (Anime) Series Watch Order:
Alternate watching Future and Despair episodes, starting with Future.
The branding can be slightly confusing between Danganronpa 3 and Danganronpa V3. Danganronpa V3: Killing Harmony is the third installment of the game series (and features a brand new cast separate from DGR1/2), and Danganronpa 3, as discussed above, is the anime split into two parts (Despair/Hope) and the last episode, Hope.