Watch gallery, following details (8)
I believe the answer is:
spectate
'watch' is the definition.
(I know that spectate is a more specific form of the action the watch)
'gallery following details' is the wordplay.
'gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
'following' means one lot of letters go next to another.
'details' becomes 'spec' (I can't justify this - if you can you should give a lot more credence to this answer).
'tate' after 'spec' is 'SPECTATE'.
Can you help me to learn more?
(Other definitions for spectate that I've seen before include "Observe" , "Be an onlooker" , "Perhaps enjoy sport" , "Watch (a game, etc)" , "Be a viewer, at a sports event perhaps" .)