Is it possible to give limited access to users in Customer Portal without logging in? (.e.g. access to the homepage and and some visual force pages). When the user tries to access say a Custom Object they will be redirect to the login page first.
You can add pages that don't require login. However, these need to be custom pages. I don't think there is a way to expose standard tabs without being authenticated.
Take a look at page 41 of this guide to see how to add custom pages.
Best of luck!
A customer portal requires premium customer licenses in order to use. It would be far more appropriate to open a chatter communities portal where you've enabled the guest user login. In that scenario you could display content that doesn't need to be secured or protected in the same manner as your customer portal does.