If you setup a second Google account on your phone, any app that uses a Google account should let you swap between the two accounts.
For example, my wife and I have both signed in to our Google accounts on our android TV. If she wants to rent a movie or download an app using her own personal funds, she can select her account. If I want to use a free rental I got for owning a Chromecast or use my own personal funds, I can select my account.
Doing something like this, logging in from both accounts, you should be able to download your apps that are already tied to your existing account as well as start downloading new apps and connecting them to the new account, or however you want to do it.
If you've PAID for apps using the "old" account, there isn't really any way to transfer those to the new account that I'm aware of.
If your biggest concern is that you want to send and receive emails from the new account instead of the old one, then that's a bit less of an issue. I would still suggest that you keep all apps tied to the old account if you've paid for apps. This way, all of your apps and app data are in one place. You can then still send and receive emails from the new account and/or still send/receive from the old one as well. If nothing else, this will help you in transitioning to the new account completely.
If you haven't paid for any apps, you could much more easily transition to the new Google account.
Hope that makes at least some sort of sense. Because of these issues, my wife of almost 6 months now still has her maiden name as her Gmail account, because we didn't want to hassle with this stuff. Again, if she wanted to send and receive emails from an address using her new last name, that wouldn't be near as big of a deal as if she wanted her apps that she's had for years in some cases tied to a new account.