The reason to choose a DSLR is to have choice in the settings, controls, and lens to use for each situation. Really, if you can't take a great photo with a point and shoot then a DSLR won't change a thing.
You also say that budget is an issue, which would knock out the DSLR. If you're looking at a superzoom camera, like the FZ35, then you will probably want some ability to zoom in closer. To get close to the same telephoto reach of the FZ35 you'd need to buy 300mm lens. This would be the 70-300mm VR lens to go along with the kit lens. That would add $600 onto what you just spent on the camera and kit lens. It's an expensive hobby and you also have to ask yourself if you're willing to lug around a big camera everywhere.
I've been camera shopping for months and I just can't decide what to get. Money is very much an issue, so I'm agonizing over the decision. I'm inclined to get a DSLR (probably the Nikon D3000) but I worry that I'm going to become one of those people who uses this powerful, multi-feature camera in auto-mode only. And then, I'd regret spending big bucks on something so bulky.
My second choice would probably be the Lumix FZ35, and though Cnet likes it, they sound pretty negative on everything in the megazoom category (though I'm not really sure why). If I'm not going with the DSLR I at least want aperture priority and shutter priority modes.
I guess I'm just really wishing that there was some quiz I could take somewhere that would help me predict what kind of user I'm going to be.