How can I make my own home-made cola drink?

William Sonoma makes something like that which you can look into.

http://www.williams-sonoma.com/products/sodastream-fizz-sparkling-water-maker/?pkey=e%7Csoda%7C8%7Cbest%7C0%7C1%7C24%7C%7C5&cm_src=PRODUCTSEARCH||NoFacet-_-NoFacet-_-Feature_Recipe_Rule-_-

I haven't tried this but it seems to fit what you're looking for.

I would check at you local coffee shop to see if they make Italian Sodas, most do, and see if they have a cola syrup. We make Italian Sodas at home with a carbonator and different blends of syrup. The kids have never pursued a "cola" and prefer lemon and other fruit flavors. We've found some syrups that the worst ingredient is sugar and it seems to work fine.

I have had a friend make this home made Ginger Ale that self-carbonates through fermentations with positive results. The only danger is the risk of over-carbonation and a resulting explosion.

I have three suggestions for you:

Red Bull makes a cola that is more what I would expect homemade cola to taste like. It has a list of ingredients you could buy in the bulk spice section of a really good grocery store. When I make it, this is what I will be using. I would recommend tasting it as well. It has a tea like taste. (Here it is: http://en.wikipedia.org/wiki/Red_Bull_Cola)

Also, here is a how to make "OpenCola". It is a brand that has their recipe available so people can experiment with their own recipes. http://www.wikihow.com/Make-OpenCola

And finally, This American Life (on NPR) recently did a story where they allegedly revealed the secret recipe for CocaCola. http://www.thisamericanlife.org/radio-archives/episode/427/original-recipe/recipe

I would use a combination of all these methods. You will need some sort of carbonizer, or you will have to mix your own syrup concoction with soda water to get fizzy soda.