Photography involves understanding concepts of exposure, lighting, and composition which you'll need to develop into skills that you can use to produce creative results. That means you'll need some basic equipment to start learning, and then you'll need to learn some fundamentals.
1) While you can learn some things with any camera, it really helps to have a camera that allows you to have control over everything - rather than have some algorithm inside the camera making decisions for you. To this end, you'd want to have a camera that allows you to have manual control over the exposure. There's nothing wrong with having a camera with automatic controls and there's nothing wrong with taking advantage of those features. But for purposes of "learning", you'll "get" the concepts faster if every decision made is YOUR decision -- so every mistake made is YOUR mistake. If you do something wrong, but the camera attempts to compensate for it using some automatic control... well you might not learn from that experience. So while I will use automatic modes even today... I learned by using cameras entirely in manual mode.
2) Having a "manual" camera, you'll want to learn the fundamentals of "exposure". The camera (film or digital) needs a certain amount of light in order to produce an image. When the camera shutter opens, light flowing through the lens is able to reach the film or sensor. It turns out the rate at which light enters depends on the lens and the size of the aperture opening relative to the focal length of the lens (they call this the "focal ratio". You can also control how long the shutter should remain open. Lastly, you can control how much light should be needed by altering the ISO setting (if a digital camera) or using a different type of film (if it's a film camera.)
Check out the Bryan Peterson book "Understanding Exposure" or the Scott Kelby "Digital Photography" series of books (specifically the first few volumes because he's now up to 5 books in that series.)
3) Pracitce
Practice a lot. No... practice even more than that.
You could just mess around with the camera randomly (and you should) but it's especially helpful to learn some concepts and then go try to put them to use to make sure you "get it".
4) Composition: There are reasons some images just look better than others. Camera position, subject orientation, the play of one color off another, the mix of light, patterns, textures, etc. All of these contribute to the result. You could flip through a bunch of photos taken by other photographers and from time to time, something will catch YOUR eye. You want to understand WHY. Why did you want to stop and linger looking at one photo... but page-flip through other photos without spending much time on them at all? The images that made you pause have "stopping value". Study those images -- there's a reason they appealed to you (and probably appeal to other people.) You'll want to learn to dissect how those images were made so that you can learn those techniques for your own photography.
To help... you might want to pick up a copy of Michael Freeman's book "The Photographer's Eye: Composition and Design for Better Digital Photos" I don't recommend you try to learn both fundamentals of exposure AND composition at the same time. Make sure you understand "exposure" first... then move on to "composition".
5) Lighting: Lighting makes one of the biggest differences in how your images look. But it's really the interaction between light and shadow. You can't actually have good light without good shadow. Ultimately you'll want to learn how to control light -- whether you are controlling available light (including returning to a location at a time when the lighting is better) or by using reflectors -- or whether you are creating your own lighting using photo-strobes (and learning to control the light using light modifiers.) You can create a lot of drama and emotion just by altering the lighting... if you want to depict an image of joy or despair... there's a way to set up lights to convey that emotion.
You might want to pick up a copy of "Light Science and Magic: An Introduction to Photographic Lighting" by Fil Hunter, Steven Bifer, and Paul Fuqua. Also go check out the Strobist blog (strobist.blogspot.com)
You aren't going to learn all of this next week... or the week after. You can pick up the basics of exposure in a few days to a few weeks. Composition will require years of practice (but you'll develop an eye for noticing good composition in less time if you practice.) Lighting is also an area where you learn the basics quickly, but mastering it takes years.
6) Don't forget to have fun!