You can take backup of your WordPress website with various mehthds. To export or download the WordPress posts via Dashbaord, follow the below steps:
On your old site go to Tools > Export in the admin area.
You can select what you want to export. In your case "Posts." Click Download Export File
To export or upload WordPress posts on your new web host, follow the steps below:
Tools > Import > WordPress
You'll have to hit a button to install the core plugin, but you can remove it after the import.
Import the file you downloaded from the old site.
Here is a complete guide on taking WordPress backup: http://www.wpexplorer.com/backup-wordpress-site-plugins/
Some of the best and reliable web hosting companies for hosting WordPress websites, I would recommend are, Bluehost, HostSoch.in, Milesweb.