Some classic Keith Richards is always a good thing. There are 2 guitars he uses in Brown Sugar. This is the first guitar, and the easier of the two. Both guitars play pretty much the same chord progressions. The main difference is that Guitar 2 uses some classic rock n' roll riffs during the verses. Learn this guitar first, and then all of Guitar 2 will be easier to get down (we've taken it out of the audio here to avoid confusion, but you can hear it in the normal speed audio).
Most of the song uses partial chords, so feel free to thumb-mute if you need to eliminate unwanted string noise from the lower strings while you strum.
Intro - Follow this chord progression:
C major - G major - C major - F major - C major (play 4 times)
Interlude - This is best broken down into small sections. The basic chord progression is simple, however he adds/removes fingers which alters the names of the chords, so don't be put off by that:
Bb6 - Eb major
C major - C sus4 - C major
Ab major - Db major - Ab major
Eb major - Bb major
C major - Bb sus4 - C major
(repeat)
Verse - Follow this progression:
G major - G major - Pull off chords - Then alternate between C major and F major (repeat)
That's all the main riffs / song sections. The rest of the song is more of the same stuff, although he alters some verses, usually by shifting up to higher versions of the same chords. You can see that played out here.