My Domain Is Required only if you want to use custom Lightning Components. If you need to simply move to Lightning Experience, you will not need to deploy My Domain.
You must deploy My Domain in your org if you want to use Lightning components in Lightning tabs, Lightning pages, as standalone apps, as actions and action overrides, as custom Lightning page templates, or elsewhere in your org.
For your question:
Can I deploy "my domain" to users and not let them know it exists?
You can, but you should not. The first step towards deploying My Domain is to let your Users know well in advance about the change. One of the reasons is because if you have say hard coded values in your code specifically in Email Templates, VF pages, custom buttons or apps, those will break when you enable and deploy My Domain. All such URLs needs to be replaced with the new subdomain URL.
In the mean time, they can still access Salesforce through the generic domain?
Yes. Even if you deploy My Domain, Users will still be able to login using say
login.salesforce.com unless and until you do not choose to
Prevent login from https://login.salesforce.com under My Domain. It's advisable not to restrict this option until Users are fully aware of this change.
Take a look at Guidelines and Best Practices for Implementing My Domain, which should provide you with more details around enabling and deploying My Domain and actions required around it.