Dog beginning to rescue miner (7)
I believe the answer is:
collier
'rescue miner' is the definition.
'collier' can be an answer for 'miner' (collier is a kind of miner). I'm not certain of the 'rescue' bit.
'dog beginning to' is the wordplay.
'dog' becomes 'collie' (collie is a kind of dog).
'beginning to' becomes 'r' (I am not sure about this - if you are sure you should believe this answer much more).
'collie'+'r'='COLLIER'
Can you help me to learn more?
(Other definitions for collier that I've seen before include "Worker down under" , "unable to avoid contact with mine" , "Pitman" , "British coal-miner" , "Miner; coal ship" .)