A. and B. Strugatsky are not that much recognized in the western countries as Andrey Tarkovsky, but they made significant contribution to the Russian literature of XX century, and their books are still popular in the post-soviet countries. In particular, the Roadside Picnic influenced many modern sci-fi works too, including, for example, the "S.T.A.L.K.E.R." and the "Metro" video game series.
Tarkovsky's "Stalker" is quite distinctive from Strugatsky's novel both in style and plot, but the screenplay of the movie was written by Strugatsy brothers too. Basically, they made completely different work in the same setting, but following completely different narrative. Also, they included a lot of pieces from their other (lesser famous) work "The Ugly Swans" into the screenplay, which is closer to Tarkovsky's film style. For example, the Writer's monologue[2] is just a direct translation of the same monologue from The Ugly Swans.
"Solaris", another Tarkovsky's famous sci-fi movie, based on Stanislav Lem's novel of the same title, was a more controversial work. Stanislav Lem made quite critical reception of this adaptation that simplifies and distorted his novel's original ideas. However, I think that Tarkovsky's "Solaris" is still the best adaptation of this book we currently have.
Aside from this, I would recommend "Ivan's Childhood" a war movie about orphaned child, and the "Andrei Rublev" a historical movie about the famous Russian Orthdox icon painter set place in the late middle ages, and about survival of artistic people in hard times.