However, mtngal's reply made me think. Is that really possible that if all settings are the same my S3 would result in better pictures than an A300 (in a well lit scene)?
In well lit scenes, you might not see much difference. What I was referring to is the depth of field - how much is in focus and how much is out of focus. Small sensor p&s cameras will have more in focus than a dSLR, especially at larger apertures (smaller numbers). So its easier to have subjects in the foreground and background in focus, especially in lower light or when there's a large amount of distance between subjects. The other reason why your pictures might suffer is that you will be using a much heavier, larger camera and some people find them more awkward to hold still. Others (including me) find it easier, so that will depend on the person.
As far as flash goes - built in flashes aren't all that great. The stronger ones sometimes just make people look even more washed out. Best thing to do if you are going to use lots of flash is to get a separate flash unit. I'm not very good with flash - I don't know enough to use it properly and I often get unexpected results.