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.
Acrylic Paint
You'll be dazzled by the number of colors you can buy, but you'll develop more mixing skill (and save money) if you just buy these few basic colors:
1. Ultramarine Blue
2. Cadmium Yellow (or Primary Yellow)
3. Cadmium Red (or Primary Red)
4. Mars Black or Carbon Black
5. Titanium White
You can mix just about any color with these five. These were the only colors I used for my first two years.
If you have money left over, consider:
6. Yellow Ochre (a more natural yellow)
7. Burnt Umber (a rich, all-purpose brown)
8. Alizaron Crimson or Quinacridone Magenta (good for mixing more vibrant oranges and purples)
9. Pthalo Green (a bluish, vivid green, important if you are going to paint plants)
These four will make your life easier but are not necessary for starting. A good way to practice color mixing is to try mixing colors to match pictures in magazines, things around the house, etc. I used to try to match paint chips.
When I was using student quality acrylic paints, I liked the Liquitex Basics series best, but I didn't really find much difference in quality between the brands. For the professional quality paints I recommend the Golden series. As a professional artist, I use about 20-25 colors.
Watercolor Paint
For watercolor paint, the student quality paint sets are usually the way to go. Watercolor comes in tubes (you only use a tiny tiny dab each time) or dried, in a pan, like the paint sets you had as a kid. Both work fine, and which one you choose is a matter of personal preference. I've found that the pan kind is much better if you are going to be carrying your paints around with you a lot.
Get a set that has about 12 colors, and look at the color list to make sure it has black. Some kits don't, and I have no idea why. If it doesn't have black, buy a small tube of it too. You can always supplement the colors in the set with individual tubes. Unlike acrylics, you won't be using white in watercolor so you don't need that. My recommendation for cheap but good quality watercolor paints is the Winsor & Newton Cotman series, but there are some other good brands out there as well.
A neat thing about watercolors, as opposed to acrylics, is that the paints keep for decades, even when dried up. I have some watercolors that someone gave from the 1970s that are dried up, but the quality is as good as ever.
If you do know exactly what you are looking for, eBay often has great deals on paint and brushes. If you're new, it's probably a better bet to go to an art supply store where you can read the labels.