There's not a huge amount that you can do, but you may want to do the following:
"Read" unread emails
Click on "More" above your mail and then "Mark All As Read"
Click on the "Gear" and then under "Inbox Type" choose "Unread First". You can select a lot of emails by checking the box at the top of the column of emails and then "Mark as Read"
If there are emails you want to keep, you can organize them by applying labels or filters. Instructions are available on the Gmail help page.
Vocabulary:
The "Inbox' is anything that is found in the large box on the right (in any tab)
A "Label" is basically like a folder, only it can be applied to more than one thing
"Archive" or "Archiving" refers to removing things from the Inbox so that they are only accessible via the Labels in the left pane.
A "Filter" can either be temporary to view or permanent to sort, but basically takes named criteria and applies them to your mail.