Although chickens will eat most anything, it doesn't mean they should. You'll find they will probably have favorite things to eat (my girls love cracked corn which is saved for an occasional treat). Chickens don't have too may sensibilities when it comes to eating a balanced diet (much like young children). It's up to the chicken owner to help provide that balance as well as keeping them away from things that are harmful for them to eat. (Here's a list of things chickens should NOT eat.)
Feed Based on Age and Intended Use
What you feed your chickens depends on their age and whether they are egg layers or being raised to eat.
Chicks should eat chick starter if they're being raised for layers and meat bird starter for those you intend to eat. If you have both kinds of chicks mixed together, feed them all meat bird starter, which is higher in protein.
Starter food comes either with or without medication. If you choose the medication route, you'll need to stop feeding your chickens this food 4-5 months before they lay eggs or 2-4 weeks before you butcher the meat birds, to make sure the medication is completely out of their system.
After the chicks graduate from the brooder, if you have a mixed flock, it's best to separate them at this point. Feed the layers a pullet food until they're about 20 weeks old. From that point, switch them to layer food, which contains more calcium, vitamins and minerals that a laying hen will need.
For meat birds (known as 'broiler' birds), use a finisher-grower food after they leave the brooder. This food is formulated for the needs of a quick growing bird, which is how broiler chickens have been bred.
Even if you have a small city flock, consider getting your feed in 40-50 pound bags (you could split it with a friend if it feels like too much). It's WAY cheaper in the long run to buy your food in this quantity, as it's the normal size the feed store carries (if they carry smaller sizes, they've generally split and weighed the bags themselves, causing the price to rise significantly.)
Depending on what kind of chickens you have and how much you're supplementing their diet (allowing them free range time or feeding them table scraps), will determine how fast you'll use that food. I find that with a flock of 8 chickens (who get some recess time out in the yard and some fruit and vegetable scraps) a 40 lb bag lasts about a month, give or take a few days either way. (Even if you're giving them scraps, be careful not to over do it. You want to make sure they're getting enough feed to keep them healthy and laying eggs with good strong shells. Because of this, some adhere to the view that feeding them anything extra (besides what they can scrounge for themselves in the yard) isn't good for them. I personally think that treats are fine as long as they're healthy and occasional.)
Other Food Stuff
Another thing that needs to be part of a chicken's diet is grit. Since chickens don't have teeth, they need something to help them break their food down for digestion. Chickens seem to sense when they need grit (tiny pebbles and rocks), so just keeping a small bowl full for the chickens to eat at leisure works fine.
If you're raising chickens for eggs, adding an additional small bowl of oyster shell is also a good idea. This helps them with the calcium they need to produce nice, hard-shelled eggs.
One time of the year that I purpose to give my hens extra protein (in the form of tuna fish, shredded cheese, scrambled eggs, etc.) is when they're molting. Because it takes a great deal of energy for a chicken to re-grow feathers (which are 85% protein), a tiny bit of extra protein aids them during this process. A good rule of thumb for HOW MUCH extra protein is generally about a teaspoon per bird per day. Again, don't over do it.
Free Range and Fresh Food
Any time your chickens can hang out in the yard is great. There might be restrictions to allowing your chickens 'recess' time if you're raising them in the city, but chickens are healthier if they have access to bugs, worms, plants, weeds, and other things found out in the 'wild' of the backyard.
They will also happily eat your food scraps. Most of what you don't throw in the compost can be given to your girls (check here to make sure it's safe). Some chicken owners report having 'carb addicts' for hens who gobble up bread, rice and cereal as fast as they can. For some reason, I can't get my chickens to eat that kind of stuff, but they do love fruits, vegetables and clover. When out in the yard, they fight over worms pulled from the ground. I've even seen a hen leap and catch (and eat) a butterfly in mid-air.
Chickens can be fed a variety of 'treats', but again, think of them as treats, and don't over-do it. My girls love both cracked corn and bird seed mixes. I sprinkle a couple small handfuls out each day (in the winter) to give them something to scratch around in their chicken run for. They love to eat it and it gives them a bit of exercise in the process. There are also commercially made 'chicken treats' that you can buy such as mealie worms. Your chickens will love you forever if you feed them those things.
(Skip cracked corn in the summer because it raises their body temperature.)
Fresh water is super important to a chicken. They need a lot of it. Depending on the temperature, a chicken can drink from 2 to 4 cups of water per day, per chicken. Where they don't seem to be as picky in the eating department, they can be very picky about their water. They don't like it dirty or hot. Because of this, it's important to change the water frequently to insure your chickens drink as much water as they need.
In the winter, make sure the water supply isn't frozen. If it freezes, thaw it out at least twice a day. If you live in a cold region, consider getting an heated dog dish or a heated base to set your chicken waterer on.