Familiarise a Conservative with odd following (8)
I believe the answer is:
acquaint
'familiarise' is the definition.
('acquaint' can be a synonym of 'familiarise')
'a conservative with odd following' is the wordplay.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'with' says to put letters next to each other.
'odd' becomes 'quaint' (**).
'following' is a charade indicator (letters next to each other).
'a'+'c'+'quaint'='ACQUAINT'
(Other definitions for acquaint that I've seen before include "Make aware or familiar with, say the facts" , "Get familiar" , "American to introduce" , "brief" , "Get to come to know" .)