Drop in to the One Kew Road opposite Richmond station - that was the Station Hotel where AOL first eyed the Stones. They apparently still have live music there. Nearby is Richmond Athletic Ground, below Kew Gardens, site of the first Jazz and Blues Festival, where the Stones played in 63 and 64, I believe. Then head down to Eel Pie - between Twickenham and Ham. Fantastic river walk. The Eel Pie ISland Hotel where loads of bands including the Stones played is long gone, and most of the island is private, but there's a good pub right next to it. And of course The Wick, near The Roebuck pub at the top of Richmond Hill and close to the viewing terrace over the Thames - a view protected by parliament. Orleans House gallery once had a fantastic exhibition about Eel Pie, with a dansette and long players by The Birds and The Stones' debut album and you could play them - they sounded fantastic from that mono set. But that was some year ago - good place to visit, however. Finally, for Who-related Richmond, the Open Book is an excellent indie bookshop on King St, and first opened by Pete Townshend in 1978 as Magic Bus. Finally finally, in light of Sympathy for the Devil, you should know that Aleister Crowley lived at 57 Petersham Road between 1939 and 1941....
There is also the Nando's restaurant in Richmond (just near the bridge). In the 60s this was a coffee shop called Auberge where the band would hang out before/after Eel Pie gigs.
And if you walk across Richmond Bridge to St Margarets you can see Twickenham Film Studios, where the Beatles filmed Let it Be. Also, along the river there is where the Beatles did a photo shoot in 1969 (the one where they are on a rowboat...and also sitting on the font of John's white Roll's Royce).
Also in St Margarets is Marble Hill Park, where Ronnie, Riod and the rest of the Faces used to play soccer (their team was called the "Marble Hill Academicals"). Ronnie Lane used to live just near there.
"We started off playing at a rugby club just about round the corner from here, that was Richmond Athletic Grounds so it's not really far from here to there" Mick statement about playing in Twickenham
My home town is Kingston, which is about 20 minutes on the 65 or 71 bus, or I have walked it or cycled it along the River Thames many times in the past.
I think all the places to visit in Richmond area have been covered in this thread, and unfortunately most of the really interesting places in the area are long gone.
This was the original Crawdaddy Club site, but very differant now on the inside. Still have live music there at times, but a restaurant now.
Mick Avory on drums in this clip from a few years ago.