What to Paint?
Most beginners strive to make their paintings look as "real" as possible. From my own mistakes, I know it's hard to know which subjects will be easy and which will be hard until you have a lot of experience. For your first few paintings, you'll want to choose something that will challenge you, but be easy enough to give you a good experience and a finished product you are encouraged by.
A lot of art teachers recommend starting with a familiar scene, like the view from your window, or a self-portrait. While these are things you'll have access to, I actually don't recommend this tactic. The more familiar you are with the way something should look, the more critical you'll be of your painting of it. For the first few paintings, you'll be busy enough getting used to the materials. Just have fun. When you're ready to choose a project, consider this advice:
When Your Painting Isn't Going Well
I find that with nearly every one of my paintings there is a point, usually near the beginning, where it just looks terrible and I want to throw it in the garbage and start over. I think this is a hurdle that a lot of new artists never get over, so I want to discuss what to do when you're at this point, and how to get through it.
By far the biggest part of knowing when to hold 'em and when to fold 'em is experience, of course. I had to struggle through a lot of paintings to know that this, or this:

Four Beginner Painter Mistakes
And I've made them all...
1) Staying in one place while you paint
You'll be surprised how much the viewer's distance from the painting will affect how it looks. What looks blocky and uneven at a working distance may look fine from a few meters away. Think about how far away your painting will be viewed, and check it periodically from that distance. Same goes for lighting: if your painting won't normally have a bright light shining on it, make sure you check it under the light conditions it will be displayed. I often take my half-finished paintings off the easel and hang them on the wall, then sit on the couch and have a snack while I make a mental list of what to correct.
Choosing a Medium
Anyone who has gone shopping for art supplies will know that there are a wide range of choices and a lot of intimidating artsy technical terms. As a young artist with no art training, I didn't know which supplies would give me the most for my hard-earned babysitting money, what I did and didn't need, and what I should buy first if I couldn't afford to get everything at once.
People often ask me what to get if they are learning to paint on a budget, or simply don't want to commit too much money to a new hobby at the beginning. This is the advice I usually give, based on my own trial and error experience. I hope it will be useful if you are starting out.
Choosing a Medium: Watercolor vs. Acrylic
There are several different kinds of paints, and each will produce a different look. I have experience with the two most popular, watercolor and acrylic, so I will discuss only these two.
Buying Paint
New painters visiting large art stores are usually tempted by the pre-packaged beginner sets, which contain a nicely arranged selection of paints, brushes, and some extras like a pallette or brush cup. My advice is don't go for the sets, they are usually very low quality and you will end up replacing them soon anyways. For the same amount of money you can buy a few good-quality colors in single tubes and two or three versatile brushes. These are my recommendations for how to get the most out of your money.
Student Quality vs. Artist's Quality
For both watercolor and acrylic, you'll notice a vast difference in price between the "student quality" and "professional quality" aka "artist's quality" paints. Since they come in exactly the same colors (and usually the same brand), you are probably scratching your head and wondering why you should waste your money on the expensive stuff. I didn't know the difference for years and originally just bought whatever was cheapest. The difference between the two grades of paint is that they contain the same base pigment, but in different concentrations. Professional quality paint is much more concentrated. The student quality paint contains more filler, hence the cheaper price.
My advice here is for your first experience with painting, buy small quantities of the student quality paints and experiment with them. If you're still into painting when the tubes run out, buy the professional stuff on your next trip. There actually is a very big difference in quality, but you won't see a big difference in your results until you have more experience.
Caring for your Supplies
Now that you've bought your paint and brushes, you'll want to make them last. Brushes in particular will last much longer if you care for them well. As a beginner, I had no idea how to look after my brushes and I wrecked a few of them out of inexperience. Here's what I recommend, based on my own trial and error:
Paint
Your paints don't need very much care. Just make sure you cap them tightly without cross-threading so they don't dry out. If you get serious about painting, I strongly recommend getting a tube wringer to avoid wasting paint, but it's not necessary at the beginning. If you're using professional quality paints, it will pay for itself pretty quickly. Also, it's great for toothpaste tubes!
Studio Extras
If you're setting up on a budget, you should be allocating your limited funds to the supplies that will make the most difference: your paints and brushes. You should also get masking tape (hardware store type is fine), a painting surface and a few inexpensive extras.
If you have room left over for more, all the better. If not, this is my advice on things you don't need right away:
You don't need an easel You'll feel more like an artiste with an easel, but for paintings smaller than about 11 x 14, a regular desk or table works fine. For watercolor, you'll be working on a horizontal surface anyways, and an easel will make things harder. For acrylic, you can work horizontal or vertical, but many beginners find it easier to try flat first.
Palette and Masking
You should be spending most of your money on the paint, brushes and the canvas or paper, in that order. However, these are a few inexpensive extras I've discovered that will save you a lot of time. I also recommend you read my post on tempting artist's supplies that you don't need.
Masking Tape
If you've been reading this blog for long, you'll know I love masking tape. It's great for watercolor or acrylic. People always say "I can't even draw a straight line" - well, neither can I, and it doesn't matter - if you have masking tape you will never have to draw or paint a straight line again.
Cheap solution for beginners: You can buy the fancy art-store kinds, but in my opinion, the regular hardware-store type is fine for most applications. If using masking tape on paper, make sure you read my post on masking tape for tips before you start.
Choosing Brushes
Brushes are tricky to shop for; most of the time, you get what you pay for, but some types are better than others. You'll get a better feel for what you like once you have some experience. From my own mistakes, I would say err on the side of getting fewer brushes, in the best quality you can afford. Plan on spending $5-10 for each brush. This is a particular area where you'll be tempted by the large, cheap, 12-piece bristle set for beginners, but don't buy it. These brushes are not usually very good, you won't use all those sizes and you'll be much happier spending the same amount on two or three carefully selected good brushes. Nothing can frustrate you faster than bad brushes.
In general, brushes for heavy texture paints like acrylic and oil are stiffer than watercolor brushes. You can choose from a range of natural and synthetic fibers. There is a good summary of brush materials here.
I personally recommend the synthetic, or "Taklon" type, for watercolor or acrylic. I would recommend getting:
1) A large flat (or angled) brush
2) A medium round brush
3) A small round brush for details

I have owned the brush in the centre since I was six years old, and have mostly kept it out of sentimental value. However, if you're using your brushes often, you'll need to replace them often. You can extend the life of your brushes by caring for them carefully.
Canvases and Paper
As a beginner, it's difficult to decide how much to spend on your painting surface. On the one hand, your painting is only as durable as what it's painted on, and it makes sense to spend a little more money on ensuring your time and effort aren't wasted on poor quality paper. On the other hand, you might feel more free to experiment if you're not worried about wasting expensive paper or canvas. It's up to you.
I do feel that it's most important to invest in good paints and brushes, so most of your money should go towards these. At the beginning, buy small quantities of student quality canvas or paper, and as you improve, work your way up to better quality.

