Hello! Well, having (until recently - October) been in your shoes with "only a point and shoot", I would have to say a 365 project is a great idea! I am thinking about doing this myself!
In my humble opinion, the camera/equipment isn't the only consideration when taking pictures. There are also things such as composition, lighting, subject, etc. Not to mention, depending on the point and shoot, some of them are pretty sophisticated in their own right. I have taken some amazing "macro" photos with my point and shoot (as seen here):
It is all a matter of patience and perserverance (is that even spelled correctly?). Anyway, take pictures of everything - maybe a different "theme" for each day, week or month. As you go, decide what subject matter you find most appealing and, at the end of your 365, when you have your DSLR, you will be that much more prepared!
Have fun!
Christine