Friend slightly mad about beginning to ride horse (7)
I believe the answer is:
palfrey
'horse' is the definition.
(palfrey is a kind of horse)
'friend slightly mad about beginning to ride' is the wordplay.
'friend' becomes 'pal' (pal is a kind of friend).
'slightly mad' becomes 'fey' (I've seen this before).
'about' means one lot of letters goes inside another.
'beginning to' suggests taking the first letters.
The first letter of 'ride' is 'r'.
'fey' placed around 'r' is 'frey'.
'pal'+'frey'='PALFREY'
(Other definitions for palfrey that I've seen before include "Old horse" , "Saddle-horse for a lady" , "Lady's saddle-horse" .)