Rogue One: A Star Wars Story takes place between the original and prequel trilogies and looks like it has some old and new characters in it. But which films are necessary to watch before seeing Rogue One to fully understand the film and all of the characters in it?
Technically speaking, you don't need to watch any.
The setting is of course heavily related to the Original Trilogy and the Prequel Trilogy, especially A New Hope, but as a story it's self-contained and can be almost fully understood without knowing details of either of the Lucas films, even Episode IV.
Yes, A New Hope introduces some characters (as does Revenge of the Sith), but Rogue One fully explores their motivations, and as such, doesn't require them to be watched first.
This is especially true if you know the culturally bare minimum about Episode IV (e.g. who Darth Vader is, and what Empire is, what is a Rebellion and what is Death Star).
Of course, you would get much more enjoyment out of Rogue One if you have seen A New Hope (and my recommendation is to actually do a series of A New Hope=>Rogue One=>A New Hope rewatch for most enjoyment). While not required, A New Hope introduces the larger setting in detail (Empire, Rebellion, Death Star, Mon Calamari) and characters (Mon Mothma, Vader, and to an extent, Grand Moff Tarkin. I don't really count Leia as a character per se as there isn't much role for her in Rogue One; nor do I count assorted useless Easter Eggs like Pondo Baba).
Revenge of the Sith introduces some of the characters (Mon Mothma, Bail Organa) but frankly, just doesn't offer all that much useful information to make it even remotely required as prerequisite.
As a side note, as per twitter answer Pablo Hidalgo of Lucasfilm Story Group, the most critical scene from A New Hope to watch before Rogue One is the 2-minute scene where Vader, Tarkin and Imperial brass are discussing how Death Star fits into imperial geopolitics (and Vader commits workplace harrassment).
A separate consideration is spoilers.
Rogue One does not, in my opinion, spoil A New Hope at all, and thus can be watched before it.
On the other hand, being an immediate prequel, Rogue One's ending is, of course, 100% spoiled by A New Hope :(
They successfully get the plans away and to Yavin, and destroy the Death Star.
Rogue One is a very interesting movie, but if you've already seen A New Hope, and you're watching Rogue One to find out how it ends, you're wasting your time. We all know that they
Get the plans and blow up the death star, only to have another built and thus destroying that one also, saving the galaxy, the end.
If you are a Star Wars nerd, you know the big-picture ending, but it's still nice to see what happens to the new characters in Rogue One. Needless to say,
They all die. No "happily ever after" here. Move along white suburban families.
If you've never watched a Star Wars movie before and you are trying this out, I would suggest watching Rogue One before you watch A New Hope. So, for newbies, go 1,2,3,Rogue One,4,5,6,7. (1-3 are optional, for reasons that will become blatantly obvious if you try to watch them. Warning: Consume at your own risk.)
1-3 are just the long way of explaining who and how Darth Vader Became who he is, through methods of terrible late 90's, early 2000's low-budget CGI and all-star acting. Either way, I love these movies, freaking watch them.
Rogue One I'd say stands on its own and doesn't require prior viewing of other movies in the saga to understand it. Ep. I-III would provide the backstory for The Force, Darth Vader, and how the Empire came into power, but I wouldn't say it's strictly necessary to have seen them first. For someone who has seen neither Rogue One nor A New Hope, I would recommend the viewing order as Rogue One first then A New Hope as this follows the correct timeline of events. The start of A New Hope picks up seamlessly where Rogue One ends.
Rogue One comes after episode 3 & before episode 4. I suggest to watch episode 1, 2 & 3 then Rogue One & then episode 4, 5 & 6.