First thing to clear up is that you won't have a "Thunderbird account". The people at Thunderbird don't hand out email accounts to ordinary users. Despite having badges and my name on the plaque outside Mozilla HQ, I don't have a mozilla or thunderbird address myself. :-/

OK, why do you think that you want to use Thunderbird? You seem to think it's an alternative to gmail and I'm afraid I have to say that isn't the case.

You can use Thunderbird with any email service that supports the standard protocols, i.e. SMTP and either POP or IMAP. That's pretty much all of them. A very few operate MS Exchange and expect (require?) you to use the outlook mail client.

So, find yourself an alternative email service provider. I like gmx and I don't like yahoo, outlook.com or AOL. I use gmail a lot, because it has very good integration to allow me to share address books and calendars between multiple devices. Whilst all of this can be found elsewhere, it may not be easy to set up and use. So I wonder why you have decided to stop using gmail. Personally I don't like using gmail's own web sites; they have lots of irritating now-you-see-it, now-you-don't stuff that slithers to and fro. But it all works very nicely on Android phones and tablets.

You can add your gmail account to Thunderbird. Use IMAP. Then you can use Thunderbird to copy or move your messages to an alternative account.

So long as you find a provider who offers IMAP you will be able to synch your email between devices. Addresses and Calendars are a different matter, and TBH, Android/Google has this particular field pretty much sewn up.

If you have money to burn then you could look at the apple ecosystem. I dare say they have the means to share data, but not necessarily with non-apple hardware.

Thank you for kind and sensible help, Zeno. Bye. Teresa