How can I best manage the Inbox?

You may utilized several filtering/sorting techniques to mange your Inbox.

-- Use Search to find people by name, email, phone number, key words and phrases included in the incoming conversation.

-- Use Filters to:

• Change the Inbox's sort order
• Mass respond to "like" campaign conversations
• Use Tags to organize similar/same issue conversations as they are reviewed. Be specific or general it's up to you! Most importantly include an Issue drop-down option on your web page contact form so conversations come in pre tagged. 
• Include a "No Response Requested" option on your web page contact form.  Why respond to people who do not want a response, when they only want to be heard.
• Keep the team Unassigned Inbox clean by using the Assignee feature to move conversations to the Me Inbox of the user who handles that specific issue. 
• Don't forget if you are looking for a conversation you can filter by received date or date range.