You will need to obtain a V317 from the DVLA or from the trader (if he has a supply), the original Logbook for the car you are selling, a valid MOT Certifcate (if the car requires one) and a copy of the current Tax Disc. Complete where required and either post to the DVLA or take it in person to your local DVLA office. In 7 - 10 working days you will receive the Logbook back which will detail the new Registration, along with the revised Tax Disc and a certificate which will allow the trader to order the new plates. Pass the Logbook onto the trader and job done ! Oh and do send the yellow slip to the DVLA (vehicle sold to the Motor Trade), otherwise if the trader drives the car and gets a speeding or parking ticket, they will come straight to your girlfriend !) By the way I believe the current fee for placing a plate on retention is £105 (send a cheque payable to the DVLA when sending the other paperwork. Oh and more importantly your girlfriend will receive the Retention Certifiate which will allow to keep the plate until she decides to put it onto another car.
As for your brand new car the process is much the same. However the dealer in question will probably do all the work for you as it is in their interest for you to buy the car from them ! (Current fee for a Cherished Transfer is £80). You will just need to supply the Logbook, MOT Certificate and copy of the Tax Disc to the dealer. You will also need to supply a valid Insurance Certificate for the new car you are buying which shows your Private Plate. I expect the dealer will provide you with 7 days free Driveaway Cover so it should be quite straight forward. Once you receive the all the paperwork back your girlfriend can have your old car with the new Registration Number (you will need to obtain the plates), or she can use her Rentention Certificate to put her Private Plate on the car instead.
I hope this info is of help to you but do let me know if you have any other questions.
By the way I manage all the admin for a large Motor Company (5 dealerships) so can help you where needed.