To have a unique domain structure, you should redirect www to nonwww using 301 redirect as other members said too. And regarding changing your webhosting, the best way is to migrate your website content at first and when everything was ready at your new hosting, then change nameservers of your domain through the control panel of your domain, In this case your website faces least downtime. Changing hosting only changes the IP of your website and search engines understand it and its a normal procedure for almost all websites to change their webhosting. So only try to do it with lowest possible downtime.

The number of backlinks is not the only factor for determining domain authority, The quality of backlinks is much more important than the quantity of them. For example if you make thousands of backlinks from low quality websites and weblogs, it doesn't have much effect on your ranking. Search engines can detect overlinking from special domains, low quality links and etc. The best suggestion would be trying to do everything naturally. And SEO needs you to be patient. You can't expect immediate or within weeks results in many cases especially for new domains.