Learn first, upgrade second. Hand your camera to a skilled photographer, and it's a good bet that they'll take better shots than you do. If you buy a fancier camera first, there's no guarantee that your photos will be any better. Once you've learned enough to know how far you can take the camera you've got, and what specific features (or lack of) are creating a problem for you, you'll have a much better idea of what to buy when the time comes. You'll also be on a better footing to take advantage of your new camera.

None of this is to say that better equipment won't help, maybe even a lot. If your current camera is a point and shoot, moving up to a DSLR will give you a lot more control. Maybe even make it easier to learn. Many point and shoot cameras don't give you a lot of control over things like aperture and shutter speed, so you may need to find ways to trick the camera into doing what you want. Direct control will make life easier. Still, learning is free (or at least cheap), equipment is pricier.

Here's an interesting Strobist post, in which David Hobby is forced to shoot with a Buzz Lightyear digital camera. It gives you an idea of how much you can do with limited equipment.