I'm sure i'll get flamed for this but like you my RT was naff to say the least and added to that I have a regional accent. My solution...loose the dross. I kept it simple who, where and what
Who...callsign
Where...position ie 10 north or overhead bloggsville (plus altiude is nice)
What...(you want).request to land, transtion airspace...whatever
Job done. Afterwards you will undoubtably be asked for aircraft type routing and alt. You'll be told the QNH and sqwark. Possibly asked POB and asked what type of service you require.
Rather than getting into a bad habit of not asking on your initial request for the 'superficial items' I found(IMHO) that I slowly learned to fit these in as I became bored of being asked for them.
One last thing, rather than give your life story on initial contact whilst on route, only to find you are told by a buisy atc to stand by. Try opening the conversation with 'G-****' request FIS, it cuts to the chase and gives you a bit of breathing space to get the rest in later.
As Aerbabe has said though your instructor is your first port of call, he's the one you will listen to the most on the RT.
A transeiver/airband scanner is great too. An airband scanner is cheap, particularly from communication specialist outfits and lets you listen to both good and bad.