I find myself giving out the same advice over and over, so I figured I'd put it all here in hopes that people will find it helpful.

General Art Advice

Don't Draw on Lined Paper

Lined paper tells me that someone didn't care enough about a drawing to find suitable materials. I recommend picking up a sketchbook and carrying it with you everywhere- it's a great way to make sure you have good quality paper around whenever the desire to draw strikes you.
You can take class notes on unlined paper, but you can't make good drawings on lined paper.

Learn to Document Work Properly

Check out the tutorial here.

This includes knowing how to crop- If a work is done in a spiral bound sketchbook, the spiral adds nothing to your piece. Even the simplest visual editing software can crop- including MS Paint.

MS Paint and Other Programs

Speaking of Paint, it has minimal functionality. A lot of beginners use it because it's free, but unless you do nothing but pixel art it will be painful to use. I tortured myself for years using nothing but MS Pain and a crappy free editing program that had some limited filters. Don't do that to yourself- download some of these lovely and free programs.

If you have money I recommend ArtRage for painting (it imitates traditional media beautifully) and Photoshop for general raster editing.  Adobe's entire Creative Suite lineup is quite nice, though I don't like the monthly payment format they use now.

Learn Basic Drawing Skills

There is nothing that you will find more useful as an artist. We've got some tutorials here, and there are plenty more online you can find by googling. Art classes will serve you even better.

I particularly recommend studying perspective, shading, line work, and composition. When you've mastered those (and it takes a while), I recommend that you study some color theory, figure drawing, and art history.

Draw from Life

If you want to learn to draw realistically, I strongly recommend starting with observation. Draw objects, people, landscapes from life until you truly understand their structures and how light interacts with them. Drawing from your mind is wonderful- -don't give it up- but it won't be convincing unless you have the skill with which to make it believable.

Learn Anatomy

Everyone knows people have a set number of limbs and digits, but there's a whole system of proportions most artists learn before being able to depict the human body realistically that you might find useful. Also it's important to learn bone and muscle systems, so that when you draw people you understand what to look for and put in.

Brief lesson, for example:
For example (assuming someone is standing up straight and facing forward, arms down), the human body is approximately 7.5 heads tall (although many people choose to draw them 8 heads), and the wrists and the bottom of the groin are approximately halfway up the body. Knees are a quarter of the way up. For faces, the eyes are halfway down the head. The nose ends halfway between the eyes and the chin. Split the area between the nose and chin into three. I'd put the lips on both sides of a third of the way down, and the narrowing of the chin on the other third.
The face should be five eyes across, with an eye's width of space fitting between the two actual eyes. Assuming the mouth is relaxed and not smiling or frowning, the corners of the lips should line up with center of the eyes.

Keep in mind different artists use slightly different systems, and that human bodies vary a little bit.

Don't be Discouraged Just Because Everything You Make Looks Like Crap

Everyone goes through this phase, and it sucks. No one is born with the ability to make excellent work off the bat- art is like any other skill set, and it takes a lot of practice and studying to hone.

It's hard to believe you're an artist when you aren't proud of anything you've made, but I'm of the opinion that people only need two things to earn the label: the imagination to envision art, and the drive to try and create those mental forms. If you can develop compositions in your mind and you are willing to try and make them real, you are an artist regardless of whether or not you are successful. That success will take practice and practice takes believing in yourself.

Draw Your Own Characters, or Make Other People's Characters Yours

Fan art is rarely better than the originals, and it's hard give people full respect for an image when they're imitating a style developed by another artist. If you are going to draw another person's character, make sure you do it your way. Put your own twist on the scene, so that it's clear that you are creating something new out of someone else's ideas rather than just reheating leftovers.

Get a Tablet for Digital Work

If you really want to dedicate yourself to the digital side of art, you will probably want a tablet and pen at some point. Wacom makes the good ones- other brands sell much cheaper versions, but at least in my experience (I bought the cheapest one I could find about five or six years ago), the cheap versions aren't even worth trying to use. I'm not pulling from a large sample there though. This will enable you to make nice clean lines and control your piece much more easily.

There are artists who manage to draw beautifully (and animate) with a mouse or even a touchpad, but they are crazy mofos and I don't know how or why they do that to themselves (I. have. tried. so. hard.).

Tips for NG People in Particular

Scouting System

The scouting system can be confusing to beginners, so check out Ornery's Handy Dandy Scouting Guide if you haven't seen it before.

Start an Art Thread

This is great for if you're looking for more feedback than you're getting in the portal. Please mention in the thread title if you may be posting NSFW content.

Art Chat
Historically the Newgrounds art community had three parts- the portal, the forum, and the chat. We used to chat over tinychat (and stickam before that, and paltalk before that), but don't have a devoted chatroom running these days.  However if you want to get something started for an immediate conversation I'd wager you could invite folks to chat in the Art Forum Lounge (also a great thread for other open discussion), you could make a new forum thread, or use the dedicated NG Art Forum facebook group to get more realtime discussions going.

A Note on Commissioning Artwork
Stop, Collabinate and listen!

Generally if you want something made it's best to find an artist (ideally on the collabinator) that suits you and PM them. If you can't pay, say so at first as a courtesy so they know what they're getting into.  Also, it might help to know how artist commissions usually function. The below system is set up to protect both parties from loss of time/money.

A standard formal art commission might involve an advance of say 1/3 to 1/2 of the overall pay, which is nonrefundable in recognition of the time, effort, and materials an artist puts in to a piece. Commissioners should be familiar with an artist's body of work, so that they won't be surprised by what they get. A contract is an excellent idea, to avoid "he said-she said" arguments later. If the artist shows the commissioner frequent works-in-progress, unpalatable results can be avoided with minimal wasting of time.