"Rearm" means an extension of the trial period, accomplished by typing slmgr.vbs -rearm
into the start menu search box. This will extend your trial period by 30 days. You have one of those left.
Then you go into limited mode, where the system will force you to upgrade. At that point you can either purchase an upgrade version and do an upgrade install or purchase a full version and do a clean install (recommended).
Either way, back up all your data or move it to a separate hard disk if you have not already done so (external disks preferred) before 05 July 2010.
You should be able to upgrade to Home Premium, but a clean install is always preferred.