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You can use SMS backup + which is a free download from the Play Store.
Alternatvely if your new handset is a Samsung, it will often prompt to transfer text messages as part of the setup process.
Android now has a built in transfer tool, which you ideally use at first set-up. Look for it in the main settings menu. Implementations differ from one manufacturer to another and, if two different ones are involved, you may only get the basic options. It's worth checking as, if it works fully, you can make your new phone pretty much a clone of your old one.
If you aren't using Gmail, have a look at the Android transfer tool. When I used it (admittedly between two HTCs), there were quite a few selectable options.
Yes I'm using gmail, and I have got all my contacts of the old Android phone to my new phone.
I now need to transfer contacts from a Windows 10 phone to the Android is that going to be difficult.
Glad to hear you're sorted. Apart from proprietary cables/apps, one of the easiest ways of transferring contacts between devices, especially if you don't have a sync'd service, is via the Export/Import function. If you choose vCard as the output format, most contacts apps can read that.
FWIW, when I briefly used a Windows phone, I found that it wouldn't display any of the fields I'd given custom names to in Gmail.
Of course, there is. Actually, with Google account or mobile phone transfer tool, we can directly transfer text messages from Android to Android. About the details of Google account, you can search online. And about the mobile phone transfer tool, just do it like this:
1. Download and install program to computer.
2. Start it and link both of Android phone to computer via USB Cables.
3. Preview and choose all text messages you want to transfer between them.
Of course, you can transfer text messages from Android phone to Android phone.
A SMS backup tool is needed, or MobiKin mobile transfer tool, and then you can back up all the SMS in a easy way.