I've been playing with root beer extracts, so will chime in.
There are several varieties. Cheapest and easiest to find is McCormick's. I found it at Walmart in the spice and extract section. I also picked up Rainbow root beer extract from my LHBS. I plan on trying Gnome and Homemade Soda Company extracts (birthday coming up, so I'm putting in my orders!).
For basic info on cleaning and sanitizing equipment, using yeast for carbonation and making soda from scratch - Steven Creswell's book is great. But if you just want to get started, you can simply pick up some extract, some sugar, some yeast, and get started. Some people recommend champagne yeasts, others recommend using Ale yeast. You want to stay away from baking yeast. I used Nottingham Ale yeast with good results several times.
If you want to try different amounts of sugars and add in different flavors, you might want to try starting out with 2 liter bottles or one gallon batches. I've been doing one gallon mixes.
I use a large pot and a thermometer - but you don't really need the thermometer. I bring 4 quarts of water up to 180 degrees and add in sugar. You can use white cane sugar, light brown sugar, dark brown sugar, turbinado, honey, ... Or you could blend your sugars. You mix until well dissolved.
The 180 degree temp will kill your yeast, so you need to get your mix back down to lukewarm before adding yeast. I put a lid on my pot, put it in the sink and run cold water outside the pot to do a quick heat exchange. Get your yeast going in a bowl with a 1/4 cup of lukewarm water. It just takes 5 minutes or so. You don't want to put your extract in when you are bringing your water and sugar up to temperature. You will lose the flavors. Add your extract after the water and sugar gets back down to lukewarm.
You don't need to do this step - but I use a funnel to pour my rootbeer into a one gallon glass jug, then I add the yeast. This lets me cap it and shake it up well to mix the yeast in.
Then I use a funnel and pour the root beer mix into 12 ounce bottles. I use 10 glass bottles, and one plastic bottle so I can check carbonation daily. After the plastic bottle feels firm, you put all of the bottles in the refrigerator. Give it a day or two for the yeast to settle out.
To keep root beer from getting all over the place, you could put your bottles of root beer in a cooler while carbing them up. That way if they explode - you don't have a huge mess to clean up (someone else on the forum recommended that - and I've been doing that).
Equipment? A good size pot. Something to measure out a gallon of water. A measuring cup. Measuring spoons. A funnel. Just basic kitchen supplies. A one gallon glass jug with lid is good to have.
Bottles? Either PET bottles, or re-used plastic soda bottles, or glass bottles with a capper. I just sanitize the whole mess with water and a very small amount of bleach - then rinse well.
I'm sure there are other ways of carbing (like a soda siphon and CO2 cartridge) that don't involve a keg. I haven't messed with them yet.