Preface: I don’t know anything about Hostmonster but some shared hosts may just not work with Bedrock. Depends on what kind of access they give you.
I think the best way to move an existing site into Bedrock is to download it into a new separate folder first. Then keep another copy of your project and start moving over files into the correct locations (like moving a theme from old-project/wp-content/themes/ to bedrock-site/app/themes etc.
Start with themes and any custom plugins. Any other plugins that aren’t yours, try and move them into composer.json and let Composer manage them.
Then you’ll need to do some manual work and update the Bedrock configs and merge in any custom settings you had before.