Choosing Brushes
A good brush is important for getting good results, and I recommend you buy the best quality you can afford. Brushes are tricky to shop for. Most of the time, you get what you pay for, but some types are better than others. Plan on spending at least $5-10 for each brush. 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. When starting, I would recommend getting:
1) A medium flat (or angled) brush
2) A medium round brush
3) A small round brush for details
Brushes in the store usually come with a temporary stiffening substance on the hair. This is to protect the brush during shipping, and it will wash out with water the first time you get it wet. If you touch the bristles in the store and the brush feels hard as a rock, this is why. Ask the person at the store to help you test a brush. Usually these people are artists themselves and can give you really good advice. If the brush does not have stiffener on it, check to make sure the bristles spring back when you push them with your finger. If the store allows it, dip the brush in some water, shake the water off, and see if the bristles stay in a clear point. If the tip splits or doesn't retain its shape, it's not a very good brush. There are many brushes out there, and eventually you'll find what works for you. These are the brushes I use daily as a professional artist:
Angle Shader
Synthetic/Bristle Mixture
By far my favorite, I use this brush for every single painting I do—often for 90% of the work. The angle brush is essentially just a flat brush cut on an angle. You can use it flat side down for wide flat strokes, or turn it 90 degrees and use it edge on for tight corners. It's very similar to a flat brush, but less common. I get the Princeton Best series and I usually order 5 at a time since they're hard to find in stores. These are very robust brushes, but after a few moths of heavy use with acrylics they start to get fuzzy and the tips spilt. Here's a three month old brush next to a brand new brush:
Round, Medium and Small
Synthetic Mongoose
The classic paintbrush: if you can only afford one, get a medium sized round brush. Look for one that tapers nicely to a point and retains its shape after it gets wet. I use the medium (size 6) brush for most of my fine detail brush, and the small (size 0) brush only for very small detail such as eyelashes, whiskers, and signing the painting at the end. Don't make the mistake of using too small a brush, beginners tend to go for the small brushes but sometimes a larger brush is better for keeping you on track. A small brush can just exhaust you before you get your painting finished.
Big Filbert
White Nylon Synthetic
I use this 1" filbert brush for underpaintings, skies, and areas where I need a lot of color at once, and a brush big enough to spread it around well before it dries. I also like to use the filbert to make puffy clouds, as the rounded tip makes nice little semicircles when tapped on the canvas. The brush in the link above is not the same brand I use, but it is a similar type and quality. My brush in the picture has lasted about a year and will be ready for replacement soon.
Rake Brush (aka Comb)
Gold Taklon Synthetic
This is a specialty brush that I don't use very often, but is excellent for specific applications where I need to make groups of the tiniest lines possible, such as fur or wood grain. Rake brushes, like angle shaders, are difficult to find so I buy them in bulk. My favorite is the Royal Langnickel with the comfort grip and plastic handle, and they are available at the Michaels craft store.
Big Flat Brush
Golden Taklon Synthetic
I occasionally use a large flat brush for applying protective varnish and sometimes for painting skies in very big paintings. I recommend the Blick Mega Brush series, which are intended for large paintings and murals and are excellent quality for the price.
Caring for Brushes
Cleaning: After painting, wash your brushes out well under the tap, making sure you get the paint out of the middle. Rub the bristles fairly hard with your fingers. Use a wee bit of mild soap if necessary. The brush hairs will get stained over time, but this won't affect their use. I'm embarrassed to say I bleached my brushes one time when I was a beginner, which caused all the hairs to dissolve. Don't do this - just wash your brushes well after every painting session and they'll be fine. It's worth it to get a small cake of brush cleaner to clean and reshape the bristles, but not essential if you're on a tight budget.
If your brush dries with acrylic paint on it, the brush will be ruined - you won't be able to get the paint out once it has dried. Leaving brushes in the jar of water will prevent them from drying out, but isn't much better for them. Never never never leave your brush in the water overnight. Sitting in the water too long can warp the bristles and waterlog the handle, causing the ferrule (the metal sleeve around the end) to loosen when it dries out. It's best to take a minute or two and wash your brushes out after every painting session, reshape the hairs with your fingers, and let the brush dry horizontally.
Storage: After cleaning, when you store your brushes, make sure they have ventilation to dry and the tips are protected. A typical pencil case (and even some brush holders I've seen for sale) won't work very well, because the brush tips can get bent against the end and dry crooked. I recommend just rolling your brushes in a rag, making sure to cover the ends, and secure an elastic band.
Replacement: It's up to you when you want to replace your brush. When brushes start to get old, the hairs don't form a good point, and the tip tends to split into clumps or get fuzzy, causing streaks in your brush strokes. If this is bothering you, it's probably a good idea to get a new brush. As a professional artist, I replace my brushes about every two months, but if you're a hobby painter you can use them much longer. When your brushes get old, I recommend donating them to a local elementary school if possible. They will make great sign painting brushes for kids.