Welcome to the wonderful world of cider. Just as @porkchop alluded to above, there's nothing easier on the whole planet to ferment. All it requires really is a bit of patience... perhaps more patience than most people have! But if you can be patient, you'll be rewarded!
One key to really good cider, in my opinion, is to use the best juice you can find, and with no preservatives. I get mine from a local orchard in October and only make it once a year. Kits and store-bought juice turn out okay, but if you use the real stuff fresh pressed from an orchard, you will be able to tell the difference.
I pasteurize my juice. Some poo-poo this practice but I just prefer to kill nasty wild things in the juice this way instead of using chemicals like Campden (sulfite). It's worked best for me in my own experience. I heat my juice to about 160 F for about 10-15 minutes, then cool and add the yeast. Pretty simple, and there's no live critters in there for sure after that. And no added sulfur (sulfite) to harm your yeast later. Yeast doesn't like sulfur, that's why they spit it out during fermentation! And that's actually a good point...
Don't be alarmed if you smell sulfur during fermentation. It's normal. After about 6 weeks, if it still smells or tastes like sulfur, that's a little bit of a concern. But, it probably won't. Give it time. Sulfur will usually disappear in the first few weeks. Rarely it takes months. But time does heal nearly all wounds. Patience and time. Nutrients can also help, but in my experience are really not required. Give it time, and don't hurt your yeast with added chemicals. My opinions.
Another main key to great cider is the yeast. You'll see recommendations all over the web for different yeasts. Most any yeast will turn out a fine cider, but they do all turn out different. My own personal favorites are Cote des Blancs (a wine yeast) and US-05. Notty is good too but can add a peachy flavor. Lots of people everywhere just rave about S-04, but for me it was my least favorite yeast I ever used -- perhaps I need to give it another chance!? Liquid yeasts of various sorts will pretty much all turn out fine and someplace in between. You cannot go wrong with good ole Cote des Blancs or US-05 in my experience.
I always recommend fermenting as cool and as slow as you can. I like to ferment at 55 F and at this temperature it will take at least 6-8 weeks to finish fermenting. Why so cool? It helps preserve more of the aromatic flavor and aroma of the apples. If you ferment warm at room temperature or whatever, a lot of this appley character is lost as the faster bubbling CO2 scrubs it out. A properly fermenting cider in my opinion is a very slow fizz that you can barely see. There should be no krausen after the first few days in the fermenter, just a super slow fizz.
For a sweeter cider, I like to rack the cider 2 or 3 times over the course of several weeks. After about the first 7 days, rack it the first time. When specific gravity reaches around 1.015, rack it again. The continual racking removes like 95% of the yeast and slows things down, making the remaining yeast really tired and helping them to stall out early, for a completely naturally sweet cider. My goal is to hit about 1.010 for a semi-sweet cider. When it gets to this point, you can also slam with gelatin and ice cold temperatures to remove even more of the yeast and stall it out. But then don't bottle right away necessarily. Give it a couple more weeks in cold temps to keep fermentation from taking off again. Admittedly, if you're not as lazy as I am, bottling at this point can be a little dangerous. If you do bottle too early, your bottles can explode if they warm up and fermentation begins again. On the other hand, if you can keg the cider then you should be all good to go, no explosions possible there! And if you're really lazy (like my current cider which has been sitting around waiting to be bottled for 7 months now!), then there's really no worries of the yeast waking up again!
Alternatively, if you're in more of a hurry to get your cider packaged and ready to drink, you can try different non-fermentable sweeteners if you want a sweeter cider. Lactose is probably the best option. I have used Splenda in the past and the result just tasted too fakely sweet to my palate, and judges in competitions noticed the fakeness as well. So I really would only suggest lactose, which is very mild in sweetness. You could play around with aspartame or other things as well, but personally I know I would notice. Some people might not mind it though, diet soda drinkers or the like. I'm not much for chemicals.
Otherwise, of course, you can drink the cider unsweetened, which is just as tasty! It can tend to the tart side though. Cider is quite tart when all the sugar is fermented away. But a lot of folks enjoy it just fine that way. My current cider (in that fermenter for 7 months) is semi-dry but tastes great and I'll be able to drink it just fine. I got too busy during fermentation and it went down farther than I'd planned, and I'm okay with it. Better luck next season, I keep telling myself! Gosh, I need to get that stuff bottled already!
More about yeast: I do find that Cote des Blancs results in a slightly sweeter tasting cider than some other yeasts, like champagne yeast which will ferment down to bone-dryness and leave just zero sugar left. I haven't used a ton of ale yeasts but some of the English and Scottish yeasts especially should leave more sweetness. Maybe try Windsor ale yeast, that one is a poor attenuator in beer, which might be really nice for a sweeter cider. I haven't tried it yet but someday probably will. Finally I should mention: A lot of people try Belgian or hefeweizen yeasts in cider, expecting crazy phenolics or fruity estery flavors. But sorry, it doesn't work that way in cider compared to how they work in beer. Just doesn't work that way. You'll still make good cider with these yeasts, but don't expect anything more special than your typical appley flavors. Just sayin'.
As you can see, I could talk all day about cider. If you'd rather read something in print, I do like Ben Watson's book Cider Hard & Sweet. It provides a very good introduction into the whole thing, and he'll provide the standard opinions on how to do everything which are all very different from my own advice, but I'll maintain that neither of us is right or wrong, just different. We're all making good cider. Overall it's nearly impossible to screw up cider! Seriously.
One final editorial comment from yours truly: I really don't add hardly anything to my ciders. No sugars, no spices, no pectinase, no acid, no sorbate, no sulfite, no nutients, no nothing. Just juice, and yeast, and maybe some gelatin to knock out the yeast at the end of fermentation. On very rare occasions I'll play around with the idea of adding a little tannin, but usually I don't. A handful or three of crabapples, if you can get some and juice them, will add plenty of tannin in most cases if you want to play with that. But that's about it. Keep it simple, use the best fresh ingredients (juice!) you can find, and you're likely to get something really good. And if you can juice your own apples, well... don't get me started on apple varieties now!
But seriously. Cider is sooooooo easy to make, and it's sooooooo delicious. Everyone should try it. I mean, gosh, you can leave a gallon in the back of your fridge for 3 months and end up with a reasonably tasty cider. I know this because I've done it several times! It works! Apple juice just really really wants to be cider, even without any assistance from us silly humans!
@voltron, I'm sure your kit will make a decent cider on your first try out, and that's cool, that's great. Enjoy it. But if you really want to get fancy about it, maybe next time you can get to an orchard, buy a couple gallons of theirs, either fresh or frozen is fine, try to get it without any preservatives, and use that. You might find yourself very pleasantly surprised with just how awesome it turns out.