I just recently got my 360 premium and I'm a little confused.  If I dont' have my xbox 360 connected to the internet and I"m signed on to my gamertag can I still get achievement points on my gamertag?? Or do I have to be online to get achievement points?? 
sonofdabeach2

You can, just make sure you create at a minimun a Silver account so that your profile can be used both offline and online and you get to keep all your achievements even online.