You should give your two weeks notice, always***. Whether or not you give more notice once you have a job lined up really depends on:
1) How you expect your employer to react and/or how she has reacted to other employees leaving.
2) Your personal situation, i.e. if you tried to give a month's notice and you were immediately escorted out, would you be totally screwed, financially?
When I left my most recent job to go back to school, I gave about six weeks notice, but I knew that my boss would be happy for me and I knew that if they showed me the door, my insurance would carry through the end of the month when my school insurance picked up.
As far as going to them before you are even sure you will leave: well, why are you leaving? Lots of times, people resign from jobs for reasons that can not be remedied. If you feel there is no room for growth, you can probably assess if that's even something that can be addressed. Sure, I believe in talking to my manager and asking for things like better projects, more money, more flexible hours, etc., but usually when people leave, there is a reason that goes beyond what the current company can provide.
BTW, I think that Alison Green's blog over at
http://www.askamanager.org/ is a great one, and she regularly addresses questions like these.
ETA: ***Not if you are being harassed or something like that, but *almost* always.
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