Pursuing first stranger and speaking (8)
I believe the answer is:
stalking
'pursuing' is the definition.
(stalking is a kind of pursuing)
'first stranger and speaking' is the wordplay.
'first stranger' becomes 's' (1st letter of 'stranger').
'and' means one lot of letters go next to another.
'speaking' becomes 'talking' (similar in meaning).
's'+'talking'='STALKING'
(Other definitions for stalking that I've seen before include "After game" , "providing unwanted attention?" , "Following stealthily" , "Carefully following" , "Covertly following" .)