I've made a hundred gallons or more in my tiny faux-"winery" over the years. I try to make wines from fresh fruit, and unfortunately good grapes are hard to come by in most parts of the world unless you're ok with French-American hybrids. Blackberry can make some great wines of similar complexity to a simple vitis vinifera variety. I would recommend that you avoid the kits and find fresh fruit if at all possible. You can buy suitable grapes for wine that are flash-frozen from vineyards around the country if you're not in a region where you can simply go to a vineyard and buy them yourself. A lot of wineries that you see in the store (sometimes called a "garagiste") make wine this way since they typically have ready access to fresh grapes. You'll need the following at a minimum:
A couple of carboys
A fermentation vessel
Siphoning hoses
Yeast
Some kind of juice that has fermentable sugars. This could be literally anything.
Bottles if you want to actually bottle the wine. You could always leave it in the carboys.
That's what immediately comes to mind. There are other things like sulfites, fining solutions (egg whites, clay, etc.) depending on the style of wine you wish to make. You might choose to not even use them at all.
I would highly, highly recommend
this book if you really want to get started. It's not a simple Winemaking For Dummies style of book, so it will delve a bit into the chemistry involved with winemaking, but it's important so that you can recognize faults in the wine at an early stage and know how to address it.
It's a great hobby if you're really into wine. You'll develop a whole new appreciation of wine, because when you taste something you'll be able to truly understand why it tastes that way. This could be anything from the oak, the yeast or the fining solution used (or not used).
Quick solution: Find a brew shop in town and ask them

They'll get you started.