Most people keep the old passport until it expires. You have to buy your air tickets in the name on the passport. It's also a good idea to carry your marriage certificate.

You can usually use an unexpired visa (of certain kinds) in an old, cancelled passport accompanied by a new passport. Many people do that with ILR stamps in order to avoid having to pay for a new stamp in the new passport, but it can be unwieldy.

I have a daughter in law who continues to use her passport in her maiden name. The banks insist on maintaining the account in that name too, even though she was married in the UK and has a UK marriage certificate.