I do a lot of wordpress websites/templates and i always develop them on a local server (WAMP) and then migrate them to a live server, this is essentially the same process you require. i have tried many available tools to date and by far the best one is the following plugin, works like a charm and very simple to use
https://wordpress.org/plugins/duplicator/
alot quicker than a manual backup and just as good, hope this helps
or..
phpmyadmin, dump the database to sql
copy the files to new server
phpmyadmin on new server, import to DB from that dump
go to config.php on new site, change db name and credentials to match
done
Might not be as simple as using a plugin, but might be simpler - depends on the host(s) and your luck