Before considering the style of pump you want, determine what type of tube valve your bike uses. Valves come in two primary varieties -- Presta and Schrader -- and the pump head for one is not compatible with the other. You can purchase a pump that is compatible with both types, but just beware of the difference.
Begin with a quality floor pump, which is the most reliable, easiest pump you'll buy. Look for a pump with a pressure gauge so you can easily monitor your tire pressure. Ideally, spend the extra money on a quality pump with a metal cylinder and parts, rather than a cheaper plastic pump. A metal pump head is also ideal when you can find it.
A floor pump is ample if it's for a child who will just be biking around the immediate neighborhood; however, if you intend to take longer rides away from home, you should also carry a frame pump. A frame pump is a small, portable pump that mounts to your frame and can fix a flat tire on the road or trail. These pumps are not nearly as quick and efficient as floor pumps, so they don't make a good primary, but are an ideal solution for the inevitable flat tire on a ride. When possible, purchase a pump with a built-in gauge.
A mini-pump is an alternative to the frame pump. These are smaller, lighter pumps that are ideal for bikes that can't accommodate frame pumps or for cyclists looking to save weight. Some are even small enough to fit in a bike pack. On the downside, these pumps are slower and more difficult to use than frame pumps.
Check the maximum pressure output of the pump and make sure it is capable of supplying the pressures you need. Check the side of your tire for specific pressure and be sure the pump you purchase is correctly matched.