I installed this app here (https://play.google.com/store/apps/details?id=com.vcard.android) and exported all my contact data and groups to a single .vcf file, afterwards I mailed it to my mail account and then downloaded it on my new phone and there imported it with the same app which I used for the export. This did the trick for me.

I installed Contact Sync and exported all my contact data and groups to a single .vcf file, afterwards I mailed it to my mail account and then downloaded it on my new phone and there imported it with the same app which I used for the export. This did the trick for me.