So you don't have a big budget or much time, and there's no local homebrew supplier in your town. How do you get what you need to make a simple but tasty cider? It's not hard! This is my go-to contribution to parties; everyone is impressed, but it's cheap and easy and you can do it too. (This costs less than $4-8/gallon, which is $ 0.5 - 1.0/pint.) You will need:

One gallon of the store brand apple juice. Be sure that the cider does not have any preservatives (sodium benzoate or sulfites). The yeast will fail to grow if there are preservatives.
One (12 oz) can of frozen white grape juice or frozen apple juice concentrate. Again, be sure that there are no preservatives on the ingredients list.
One packet of yeast (I recommend ordering this one: "cider house select cider yeast" because the yeast falls to the bottom and clumps together when it is done, so you can pour the cider out of the bottle when it is done brewing, and the yeast will stay behind.)
Optional: 1 tablespoon of black-strap molasses or 1/2 C of dark brown sugar (which is basically white sugar plus molasses). The molasses really helps add some afterbite and silkiness to the cider that compensates for using the cheap juice instead of nice orchard-pressed stuff. You can leave it out, but I think you'll find that it is a nice touch.
An empty 1-gallon bottle or pitcher to serve the cider from.
Space in your refrigerator to place the pitcher/bottle of cider after you brew it, until you drink it. It won't last too long, so you may want to keep some friends on hand to help you finish it.
One week of time to brew the cider. I recommend starting to brew 7-8 days before you want to have the cider available.

Step 1: Step 1: Brew Day

Once you have all of your ingredients, do the following:

Let the frozen concentrate thaw.
Pour out about 2 cups of juice from the 1 gallon bottle of juice.
Pour the thawed concentrate into the gallon bottle.
Pour the molasses or spoon the brown sugar into the bottle.
Place the lid on the bottle and shake it up to mix.
Open the bottle, and pour about 1/5th of the yeast from the yeast packet into the gallon bottle.
Place the lid loosely (!!!) on top of the bottle, and leave it in a cool / dark place (closet or basement). Don't tighten the lid or the bottle may explode once the fermentation begins.

Step 2: Days 2-7: Monitor Fermentation

This is the fun part! Sit back and let the yeast do their thing. After a day, the bottle should become cloudy as the yeast grow. You will see tiny bubbles rising to the top, and maybe a little foam, but probably not much. After about 5 days, the bubbling should slow down, and the bottle will begin to look more clear. At this point, pour out a small amount of cider and take a sip. How sweet is it? If you like it very sweet, you may want to end fermentation at this point. If it is too sweet for your taste, recap the bottle (loosely!) and let it continue fermenting. Taste it again in a day or two. Once you like the sweetness level, you are ready to stop the yeast.

Step 3: Day T-1: the Final Countdown!

So you decided you like the amount of sweetness, and you want to stop the fermentation. All you need to do is place the bottle in the refrigerator! That wasn't bad, eh? Ok, so now wait a day for the cider to cool, and for the yeast to fall to the bottom.

Step 4: Day T: Party Time!

So your cider has been in the refrigerator for one day. It should be clear now, not cloudy, and not bubbling very much. It is almost ready to serve! You can actually drink it from the bottle, but if you want to improve the presentation, I recommend pouring it from the bottle into a pitcher or another 1-gallon bottle. Do this slowly and carefully, avoiding splashing the cider, so that the carbonation is not lost, and so that you do not disturb the yeast at the bottom of the bottle. There's nothing wrong with drinking the yeast, they just make the cider look a little lumpy. They will also make the cider taste bitter if they are left with the cider for a long time. You can rinse out the 1-gallon bottle and save it for putting the next batch in.

Store the cider (in the new bottle/pitcher) in the refrigerator, covered, until you drink it. You should drink it all within 2-3 days because it will eventually go bad. Remember, don't close the lid of the container too tightly, or the container may pop! The fermentation will continue in the refrigerator, albeit at a reduced pace.

Step 5: Tinker!

Now that you are hooked on making your own cider, here are some things you can try if you want to add variety:

Different juice combinations: white grape / peach, strawberry / apple, cranberry / apple, pineapple / orange, ...
Adjust the amount of molasses: try mixing several drops into a glass, tasting, and repeat. This will give you an idea what the molasses is doing to the flavor, and how much you like vs how much is too much.
Try different yeasts: I also like 'Cote des Blanc' which is a white wine yeast, and 'Lalvin EC -1118' which is a champagne yeast. They will each add a different flavor to the brew. However, the champagne yeast will brew much faster, so start tasting after 3 or 4 days rather than 5-6.