2002 Vauxhall Zafira problem pulling away when cold starting

When I start engine from cold without touching the the throttle the engine revs to just over 2000 rpm then drops instantly to 1000-1300 (idling when cold rpm which is normal ) but fluctuates slightly sounding like the engine is about to cut out if you know what I mean . I guess sounding rough is what I am trying to say.
Then within about 2mins it revs up to just before 2000 rpm drops down to idling normal . At that point you can drive away with no probs and it's fine , if you drive away without waiting for this to happen from a cold start you drive down the road with acceleration one second then no acceleration the next which annoys other drivers and looks like you can't drive . There was no fault light on And I got my friend to have a look as he has a machine to plug in. There was no fault according to the test he performed but he maybe thought that the lambda sensor needed replacing .

This has been happening for about 4-6 weeks now . At first we didn't get any lights on the dashboard but about a week ago(the day after the machine test) we got the engine shape light on . I asked my friend to plug it in again and it showed up fault "fuel trim malfunction bank 1" code p0170 . I don't know whether this is related to the cold start prob or I have another thing wrong but I started to remember about a year ago we got the engine shape light on the dashboard for about 2weeks then it went off but we didn't have an issue with starting engine from cold . Could it be I have 2 faults ? . On reading posts on here and the net I'm guessing the engine coolant temperature sensor but I am also reading it could be the mass air flow . Can't they make it simple (aaaaghhhh!) any ideas and rough prices for these parts . Please help ! !