I've done it pretty much every way you can imagine. Each way has advantages and disadvantages.
Making ginger beer is pretty easy. Grind up a piece of fresh ginger about half the size of your thumb with a cheese shredder. Seep with about 5 cups of nearly boiling water until the water returns to room temperature. Do NOT proceed until the water has cooled. Stain the mixture (or don't if you want), and pour into a 2 liter plastic pop bottle along with 6-8 oz of sugar and 1/8 tsp of active baking yeast. Fill with water(preferably filtered), leaving an inch or two of headspace. Cap tightly and leave at room temperature for a couple of days. When the bottle gets too hard to squeeze, it's ready. If you leave it out too long the bottle will eventually explode making one helluva mess not to mention scaring the shit out of everyone because it will sound like a bomb when it goes. I've never had this happen and I think it would take a week or more, but you've been warned. Once it's ready you can refrigerate to slow the fermenting process. After opening, the remaining ginger ale will continue to ferment and make more fizz. The possibility of explosion is still there, but I once forgot about a bottle in the fridge and it stayed there a month without exploding. As with other methods, you can experiment with the amount of ginger and sugar to suit your tastes. This method will produce alcohol as a by product, but in extremely low levels. You can give it to kids without worry. The disadvantages are it takes time and you will wind up with a yeast taste which is unlike commercial ginger ale. You can use other types of yeast available in home brew stores that have less taste.
One of the easiest and most convenient ways to do it is with a seltzer bottle. The disadvantages are cost and the level of fizz you get. You can't put anything besides plain water in a seltzer bottle, so by the time you thin down the seltzer with your syrup and stir, you don't wind up with a lot of fizz. Buying the CO2 chargers in bulk reduces cost, but it's still relatively expensive. You are also volume limited by the size of your seltzer bottle and it takes a little while to cool the bottle and water down in the fridge.
Lots of people use soda stream and similar products. This is also a relatively expensive method because you are somewhat obligated to using their proprietary supplies. There is a way to hook up a CO2 bottle to the soda stream to reduce costs. I've never used their products, so I can't say how well they work.
If you plan on making a lot of soda, it's worth it to buy a CO2 bottle. The only real disadvantage is startup costs. Since I make a lot of seltzer, this is the method I use. If you are interested in this, I can give you more info.
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