On your new hosting place, create a new/empty WP install. That will create a database. Remember the credentials.
Then create a new database, and import the data you got from the old host. Remember the credentials; assign the user/pass to that database.
Then go into the new install, inside the wp-config.php file, and change the database name to the new one, with the credentials that match.
Then log into the new install’s admin area. If needed, use phpMyAdmin to edit the wp-users table to change the email for the admin user, then use the lost password thing to reset that password. (You could do this via editing that record in phpMyAdmin, entering the password, changing it to MD5, etc. but the lost password thing is easier.)
That should allow you to log into the admin area of your site. Do any updates of plugins/themes (you will need to copy the plugins/themes folder from your backed up files that you got from the old host).
Since the database contains all of the posts (and configurations), specifying the new database name for the install should get you your content. (Remember that you imported your content from the backup you got.)
There might be some more tweaking involved with plugins and theme settings, but that should get you started.