Better periods after skipping British tour (8)
I believe the answer is:
outstrip
'better periods' is the definition.
'outstrip' can be an answer for 'better' (better can mean to outstrip or outdo). I'm unsure of the 'periods' bit.
'after skipping british tour' is the wordplay.
'after' is a charade indicator (letters next to each other).
'skipping british' becomes 'outs' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'tour' becomes 'trip' (I've seen this in another clue).
'outs' put next to 'trip' is 'OUTSTRIP'.
Can you help me to learn more?
(Other definitions for outstrip that I've seen before include "As to purist, surpass or run faster" , "Go faster or do better than" , "Go far ahead of" , "Move faster than and overtake" , "Surpass, do better than" .)