ive never done this before..but is it possible to bring your dog with you in a bag on a plane? not in storage but with you on your seat?

As far as I know, they will not allow people to do this unless the dogs are for medical purposes (seeing eye dogs, etc).
Alot of airports and airlines have their own websites nowadays though so it should be easy enough to find out, or to call and ask.
Where is it you are going? What type of dog is it? Pretty sure most of them require the dogs to be in a very specific type of container and checked into their "cargo" area.

I have a friend who takes here chihuahua in the cabin with her. She's never had a problem.

No, you can definitely fly with non-service dogs. I've flown with my Jack Russell terrier in a dog carrier in the cabin with me a ton of times, but not so much since 9/11. Back then, all you had to do was buy a pet ticket when you bought your own ticket ($50 or something?), and get a letter/form from the vet saying your dog is good to travel. You have to let the airline know in advance because sometimes they only allow 2 in-cabin pets per leg of the flight.

These were the rules pre-9/11; i would check with individual airlines about their policies before booking flights. For example, here is Delta's webpage about traveling with pets in-cabin.