What are the best tutorials available to learn photography?
Does a tutorial need to be specific to my camera model?
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This is nearly impossible to condense into a single question / answer. The entire site you see here, as well as countless others, is devoted to pretty much exactly the thing you describe.
You asked about learning your specific camera, and yes, you're going to have to do that. Your user's manual is a great place to start. If you have a relatively popular camera, you may very well find additional books that attempt to present the same material in a manner better suited for learning, rather than reference. If you tell us what camera you have, we might even be able to help with this; that question is probably well-suited to a question of its own.
But that's just the beginning.
Next, you're going to have to learn about photography. You're going to need to learn about exposure, lighting, composition, and all the other topics you see here that don't tie to a specific camera. There's no such thing as the best tutorial of all, because everyone learns differently. Therefore, I'd recommend soaking in whatever reading you can get your hands on (and there's a ton of it on the web) until you feel confident targeting some of those specific areas above. If you're looking for a book, anything in Amazon's "digital photography" list would get you started -- there are some great books that show up right at the top of this list.
You should plan on reading a little and practicing a lot, then repeat until you've taken a few thousand photos. At that point, you should be fairly self-sustaining (you'll have a good idea what you need to get better at). If you come up with specific questions along the way, this site is a fantastic resource for answers, of course.