Transfering Whats app account to new phone

Last week I received a message on my phone that my whats app had expired
and that I would have to download the latest version. I tried doing this
several times but received an error message that there was no space on my
phone to download the latest version.

I have now bought a new android phone (LG G4) with a new sim from my existing
provider and intend to transfer my old phone number to my new phone.

I would also like to transfer my old whats app account with records to my
new phone. Can anyone let me know how I can do this. Do I simply
download the whats app software from their site to my new phone and let it automatically
recognise my telephone number and transfer all the previous details and
records. Or do I need to do something else.

Many thanks

Richard