Can I use CiviCRM to print mailing labels etc?
While I'm as big a proponent of CiviCRM as anyone - it's worth considering why you're looking to switch if your current tool is working for you.

CiviCRM opens up many possibilities that aren't available with Access - from mass e-mailing to allowing members to sign up/renew online to offering members-only content on your site and more.  However, it won't make you an expert in membership management or plan your renewal campaign.

If you're looking to make your organization more effective by growing membership or making better use of your member data, you'll need a strategy to do so.  That might include upgrading your database.  However, organizations who adopt fancier tools without a strategy often have buyer's remorse, even for a free tool.