You'll find some good recommended hosting choices if you click on the "hosting" link at the top of this page.
As for installation, you said you're running it on your PC already ... so apparently you know how to "install" Zen Cart already.
Installation Instructions: https://www.zen-cart.com/tutorials/index.php?article=107
Moving from PC to Server: https://www.zen-cart.com/tutorials/index.php?article=99
Granted, this assumes you have some familiarity with FTP programs (like FileZilla) to transfer files to/from the server, and take some time to familiarize yourself with your hosting company's control panel in order to create your MySQL database, etc. Their FAQs should help you in that area.
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