You can't really do the installation at your web host until your nameservers settings have been changed to point at the new web host and have propagated through the internet nameservers. After that has happened, then you can set up and configure the wordpress software, so there will be a small window of time when your blog will not be accessible via your domain name.
You can, ahead of time put an index.html file in the main directory at your web host so that when things switch over, people will see an "under construction" message.
Image links in your posts/pages/sidebar will still point to your blog here ( http://childperspective.wordpress.com/ ). There are two ways to deal with this. Leave the images hosted here, leave your blog here set to public but block search engines so that your images are served from your account here, or upload all the images to the new webhost and then edit each image link in your blog (posts, pages, sidebar) to reflect the new URL for that uploaded image.
Have everything ready to go:
1. Export all your stuff from your blog here to your desktop so it is ready to import.
2. Have the OPML file URL for your blog here written down so that you can quickly import your links. It will be http://childperspective.wordpress.com/wp-links-opml.php .
3. Have the theme you are going to use downloaded to your desktop, unzipped and ready to upload to your new blog.