Work in the smallest of steps that allow you to make progress. For many of us, "zero inbox" is not a task but more of a project.
1) As someone else suggested, move everything to a folder. Get it out of your way. As new things enter your Inbox, deal with them. Respond, file it, or archive/delete it.
Personally, I file/organize virtually no email. I archive it. If it is important enough that I'll need it in the future, I will know the keywords I need to search for it. The only thing that I do "file" is unread email and I do that by automatic filters.
2) Identify one type of newsletter (coupons, offers, other). For example, Macy's weekly special. Archive it or unsubscribe. Mass search for the backlog of Macy's then mass archive/delete those as well. You're one step closer.
3) I have two types of filters for organizing email. One is mainly focused on bills and other email that I must review. But, they do not need my immediate attention so I don't want my email client, or phone, notifying me of them. So, I have filters that place bills, statements, etc. in "money".
Consider filtering out the items like bills, statements, and other must-reads to a folder. Review that folder periodically but get them out of your Inbox or the area that may require your immediate attention.
4) Newsletters, offers, etc. I filter those into their own "news" folder. Again, consider not allowing them to hit your Inbox if they do not require your immediate attention.
About every couple of weeks I will find that a newsletter never was or has become non interesting to me. I unsubscribe from it.
5) After some time, most everything the hits your Inbox are items from friends, family, co-workers that you'd like to read and/or respond to immediately.
Regardless of how often you check email, try to take an action on each one. Respond, file it, or archive it.