Right,I am an expert in these fields as I have three private number plates so here we go :
To check if a plate is available goto
http://www.dvla-som.co.uk/home/
For changing plates you either need the v5 that the plate is on or a retention certificate
Take this with your existing tax disc,V5 and mot certificate to a dvla office
You will then have 2 options
1) Wait about 45 mins and they will give you a new tax disc in your new reg,keep the V5 and send you a new one and change your mot cert
2) Walk away and they will post to you your new tax disc,V5 and new mot cert
In both cases they will give you a letter stating that your doccuments are in the hands of DVLA and you have just changed plates.This is incase you get pulled over.You can also use this letter to get plates made.
Oh and don't forget to tell the insurance
Any more questions just ask or PM me