There's loads of microbrewers out there making wine 1 gallon at a time. Unfortunately there aren't any 1 gallon kits to my knowledge.
To get started with 1 gallon is easy. The problem is that 5 bottles of wine disappears so fast when you are brewing its not even funny. I'd recommend buying a basic 5 gallon starter kit and doing a few 1 gallon experiments on the side. That's what I did getting started and I never looked back.
If you are sold on doing just 1 gallon, you will need the following at bare minimum:
Food grade open container like a pot that will hold a little more than 1 gallon.
1 gallon glass wine jug like a Carlo Rossi bottle. Or a whole foods apple cider bottle.
A properly sized rubber bung, preferably one with a hole in it for an airlock.
An airlock. I like the 3 piece kind because they are easy to clean. A lot of people here swear by the bubble airlocks.
Racking cane, food grade tubing and a bottling wand.
Hydrometer and hydrometer test tube.
Cheesecloth
Paint strainer bag or a (unused) women's stocking.
At LEAST 2 OZ Potassium Metabisulfite
A cheap spray bottle for sanitizing
A yeast packet. Be choosy, pay attention to what temperature your wine will be fermenting at and what flavor characteristics you are looking for.
2 OZ of Bentonite
Enough juice concentrate for 1 Gallon OR 12-16 LBS fruit.
Pectin enzyme if you're using fruit.
Please note: ALL of this with the exception of the fruit and 1 gallon containers is typically included in a (5 gallon) starter kit. And it usually comes out cheaper than buying everything individually. So again, starter kit plus 1 gallon wine jug is probably the way to go. This hobby is addicting. You won't use the 1 gallon jug often after you empty your first 5 wine bottles.
A great wine recipe to start with can be found at
www.skeeterpee.com. Divide portions by 5 for 1 gallon if memory serves me correctly. It's a great first wine if you don't want to do a 5 gallon kit. (Or even if you do.)
There are sponsors here who sell all this stuff. Or you can do a search for it on amazon.
Mistakes to avoid: SANITIZE EVERYTHING. Don't add anything outside of the recipe for the first 3 months. And even then do lots of bench tests before you add anything. Ask lots of questions here if you're looking to experiment. As Solomon said, "there is nothing new under the sun." If you're thinking about it, someone here has probably done it and will be happy to help steer you in the right direction.
Good luck!!!