It's a little hard to give advice not knowing what your baseline skill level is. If you could post a sketch or something it might be helpful in showing people what areas you're strong in and what areas you need to improve in.
In general though, I think it's great that you want to start developing your skills and here's the advice I would give:
1) Draw something every day
This might seem obvious but a lot of people can't commit the time to do that. That being said, if you can commit to drawing at least one thing every day (it can be anything, literally) you will see a lot of improvement over time and you will improve more quickly than if you aren't drawing every day.
2) Get comfortable drawing stick figures
For almost everything you draw, whether it's a person, animal, or a futuristic sex robot, it's going to start off as a stick figure that you "flesh out." I can't stress how important stick figures are, especially if you're drawing people, because they allow you to plan out a pose and think about what you're drawing before you start putting down heavy lines that are going to be hard to erase.
The book "Figure Drawing for All It's Worth" by Andrew Loomis is available to read online for free and I think it is the best book I've ever read on the subject. You should also check out Loomis' other book, which covers some of the same basic principles of figure drawing but applies it to cartooning.
3) Look at how other artists draw things and learn from that
People might disagree with respect to this point, but when you're learning to draw, I think it's helpful to look at artists that you like and see how they draw things. You're not necessarily copying (although I think that has its place to when you're taking your first steps), but you are looking at somebody's technique and trying to apply what they do to your own style.
For example, in the comic book I draw, I have a lot of characters who have cybernetic body parts, and in figuring out how to render that I looked at a lot of the stuff that Yukito Kishiro has drawn because I like the way he makes robotics and futuristic armor look.
4) Draw on paper - Ink on paper - Color on a computer
Don't try to learn to draw using a tablet like an Intuos or a Bamboo. It's much easier to draw something on paper than it is to draw it on a computer and right now everything you draw on paper will look better than anything you could draw in GIMP or Photoshop.
The reason for this is because there's a certain quality about lines you draw with a pencil or a pen that is not easily replicated by computer software. That hand drawn quality is what gives your art character and you want to preserve that.
In terms of pencils, you can use pretty much anything. I use a mechanical pencil with .05 mm or .07 mm lead.
For inking (tracing your pencil lines in ink) I would recommend getting a set of artist pens. You can get a set of 4 pens from Faber-Castell for about $13, and right now that's what I would go with if I were you.
That being said, absolutely learn to color things on a computer, because that's pretty much the industry standard now. If you can afford it, get Photoshop. If you can't, download GIMP for free and learn how to color using layers in that.
And once you have a better grasp on drawing, by all means draw in whichever way feels best to you. If that means you draw everything on a computer, then go for it.
5) Pay attention to criticism but place every critique into its proper perspective
When you draw, whatever you draw, you're going to make mistakes. If you post your art around the web, and certainly on Newgrounds, people are going to point out the things you did wrong. Sometimes critiques will be helpful, but sometimes they won't.
My personal advice would be to take it all with a grain of salt. Pay attention to criticims that seems helpful, and ignore the people who tell you that you suck without giving you an explanation for why they think that.
One thing to keep in mind is this: the art community does not have a hierachical structure. What I mean by that is, there's no one person or group that can "make" you either a good or a bad artist. There's simply what people like, and what people don't like. And if people like your work, then you're doing something right. And if enough people like your work, then it doesn't really matter what the "artists" think.
Figure Drawing for All It's Worth by Andrew Loomis
Fun with A Pencil by Andrew Loomis
How to Draw Comics the Marvel Way by Stan Lee and John Buscema(it goes over basics of anatomy, foreshortening, and perspective, as well as giving some general advice about visual storytelling and is written in a way that's accessible to people taking their first steps)
DeviantArt Resources and Stock Images(lots of different poses and costuming to practice drawing)