Your suggestion for an initial 'free-call' to a busy Approach Control unit gets my vote, and is preferable to a long winded intro.
"GABCD PA28 inbound" or "GABCD PA28 for transit", gives the controller the option to reply with either "pass your details" or "standby", if he/she is not able to immediately process your request. Less chance of being asked to repeat your message, and more likely to receive your clearance without delay.
When asked to pass your details, please include your point of departure, the type of clearance required, VFR/SVFR/IFR, and where you wish to enter the Control Zone, if applicable. Perhaps it's also worth mentioning that a number of Control Zones have published VFR routes/lanes, so a bit of pre-flight 'homework' is useful.
There are standing agreements at some Airports, between ATC and based operators for 'local flights', with abbreviated R/T and procedures.
If you are ever unsure or in difficulty, please seek early clarification or help on the R/T. Your enjoyable flight to Luton should hopefully be typical of ATC in general.