I was taught - and it is my current practice and has been for a long time, that if you're calling an agency that is not expecting you - "free-calling" in UK parlance then you open with simply:
Your full callsign and what you're requesting (in simple terms).
eg: G-XXXX requesting Traffic Service and Zone Transit. Depending on circumstances I might also add the point at which i wish to transit via and also whether i'm inbound or transit only. (Requesting Traffic Service and Zone entry via Blogstown inbound to the field)
That's plenty and has seemed to work for me. Complicated explanations of present position, requested route etc just confuse on initial call.
If i've listened to the arrival ATIS and it includes an instruction to report ATIS and QNH on first contact then i add that as well.
Happy to be corrected by a controller. (Although i'll probably keep doing it my way anyway!
