What is a good plugin so a user can enter a code(s) and redirect them to a new page based on that particular code?
Gravity Forms. You can add an unlimited number of "confirmations" which are displayed html, http redirects, or page redirects after submission. For each confirmation you can add "conditional logic", e.g. if a dropdown = 1 then do this, or in your case, if the text input equals the code you provided then use 'this' confirmation. So a user enters 1234 they are sent here, if they enter 1111 they are sent here, etc.
This goes over it a bit [[LINK href="http://www.gravityhelp.com/gravity-forms-1-7-new-confirmations/"]]http://www.gravityhelp.com/gravity-forms-1-7-new-confirmations/[[/LINK]]
You can also add validation logic, which would check for incorrect (not existent) codes with hooks.