country, exclusive, private
members of an exclusive club

local
social
youth
fan, sports
fitness, health
badminton, golf, etc.
She plays at the local tennis club.

chess, drama, etc.
book

belong to
She belongs to a book club.

become a member of, join
form, found, start
run
Who runs the tennis club?

chairman (esp. BrE), manager, member, official, player, secretary (esp. BrE)
basketball, cricket, football, etc.
bar (esp. BrE), house (usually clubhouse)

in a/the club
How many people are there in the club?

comedy, dance, jazz, night (usually nightclub), strip
gay, lesbian

circuit, DJ, scene
a new style of music on the New York club scene

Club is used with these nouns as the object:
seal