What is the best way to download all my Gmail email messages and attachments and retain the option to sift through them (in a user-friendly way)? Kind of like storing Gmail offline. Is there a way to do that?
Gmail, by its nature, is an online keep-all-the-information beastie. While there is a Gmail Offline Chrome Extension, that's really only to allow you to work with (some of) your email when you lack a connection to the cloud. Once you reconnect the information gets synced back up.
You can easily download attachments to Google Drive, but it doesn't keep the connection to the message, and that's just more online storage.
In order to do what you'd want, you'd need to download and install a third-party email client and use it to keep your email messages locally after removing them from your online Gmail account. Thunderbird is one I've used in the past, but there are several that would probably suit. There are probably a couple of ways you could tackle this, but likely you'd keep a separate, offline mailbox with your messages and attachments that you'll have moved from Gmail.
However, installable software is beyond the ken of this site.