First off, start looking and see what the price range in your area is. What can you expect for what you can pay? Price range is HUGE and determined mainly by where you live.
So start looking in the papers and online.
When you find a place that looks like you can afford it, go by to have a look before you contact anyone. Figure out what kind of place it will be. A super cheap rental can mean it's in a bad part of town, or there is something wrong with the place. Or it could just mean the landlord only needs that much money.
Also remember that many places charge one or two months rent as a deposit, so you'll need to have that ready.
After you've figured out what your price range can be, then you can start trying to find a place to rent.
You have two options- you can go through a company or an individual.
Again, it depends on where you are and your price range. Talk to people and check online for reviews on a rental company before you use one.
With a company, you may get a better contract. They may, for instance, be willing to pay for a new place for you to live if your current home is damaged or destroyed.
Companies may be more picky. Some will not rent to people that can't prove they can pay, or require someone to co-sign.
Renting from an individual can have perks- you may be able to waive part or all of your security deposit, work for part of rent, etc.
But renting from an individual also means you may not get treated as well.
Remember that the only thing that counts is what is in the contract. So read it carefully. Ask to see a contract BEFORE you decide whether you want the place or not.
With an individual, do not be afraid to ask about things that are not in the contract and change it as needed (with them, of course, not alone).
Things to look for:
Security deposit AND rules for getting it back- A security deposit is usually kept to cover any damage you do to the house when you move out. If you have pets, this may be nonrefundable or you may be required to pay an additional deposit, possibly for each pet.
Monthly rent and late fees- When is rent due? When are late fees, if any, applied? How many months can the rent be late?
Eviction procedures- What qualifies you for eviction? What is the process?
Breaking/extending/altering the lease- What are the procedures for leaving, if you need to break the lease? Will you be charged the full amount for your time there? How much notice do you need to give? Who is responsible for finding a new renter? What conditions allow you to immediately break the lease and leave (eg. house is unlivable due to storm, etc).
Damage to the house- who is responsible for fixing damage to the house? If the house is damaged to the point it is unsafe to live in, who pays to move you? Will they pay to put you up in a different place until it is fixed? Who can fix damage to the house? How long does the owner have to fix damage before you can legally break the lease?
Home inspection- What is the procedure for an inspection? How much notice, if any, must you be given? What are the grounds for an inspection? What passes or fails? How long do you have to fix any violations?
Pest control- who pays for it? Who is responsible for pest control? What does pest control cover? What happens if you have non-insect pests (eg rats, raccoons in the attic)?
Things to look for in the house:
Windows and doors- these should shut firmly and seal, to keep your utilities down. Make sure windows open if you want them to. Make sure none are cracked or broken.
Carpet/floor- general condition. Carpet is better insulation but harder to keep clean. Wood floors may require special care.
Electricity and outlets- this is a big one. Check ALL the outlets and make sure they work and are covered properly.
Heating/AC- if you can, look at the system. Ducts (tubes) should be intact, and shouldn't look like they are repaired with tape. Everything should fit snugly.
Individual AC/Heaters- make sure they all turn on and work. Window AC units should be tilted slightly downward, check under windows for water staining that indicates they are draining into the wall (a nono).
Walls, ceilings, etc- check for any discoloration or water damage, which indicates leaks.
Pipes, etc- look under sinks, around the edges of the tub, and anywhere water may leak from pipes. Check for discoloration or mold.
Basement- if the house has one, it should be dry. Check for mold, mildew, and standing water.
Bathrooms- make sure all fixtures work correctly and there is room for whatever you want in there. Imagine it with all your stuff in there.
Surrounding area- what is the neighborhood like? Do you feel safe?
Convenience- how close to where you work/go to school/etc is it? Can you walk or bike where you need to go? How far is it to grocery stores, etc?
Parking spaces- do you have a place to park at your house or apartment? If not, what are the rules for parking on the street?
If you have a job, ie a source of regular, provable income, and can actually afford to support yourself, then it's just a case of doing the practical thing and actually finding somewhere to live. A room in a shared house might be the best option because it'll be much less expensive than your own place. Check out local newspapers and classifieds websites for potential places to rent and take it from there.
You'll most likely have to sign a contract for a set length of time, so think carefully about whether or not you can definitely afford to support yourself for 6 months, or a year or whatever. You'll also need a security deposit saved up, which is usually at least the value of one month's rent, and you'll pretty much definitely have to pay that AND your first month's rent up front, which can add up to quite a bit of money.
It might be worth asking one of your parents if they would co-sign a lease for you. This is basically just a way for a landlord to be sure that you're going to be able to pay your rent in an emergency. I moved out of my parents house at 18 and never had to do this, but I know some people who have. It really just depends on the requirements of different landlords or rental agencies.
Also, never underestimate the expenses of general stuff and things! Consider ALL the bits and pieces you'll need to buy, things you use every day like toilet paper, soap, laundry detergent, as well as your electricity bill, internet and phone etc. And food! Spend a week making a quick note of everything you use and munch on a daily basis and then add it all up at the end to get an idea of what kind of money you'll need for living expenses. Don't forget to include a bit of contingency for emergencies, like if you need to replace a lost phone or get your computer fixed. It's definitely worth looking into contents insurance too to protect your belongings.
I left out right before I turned 18, and if there's a will there's a way. I'd suggest having a job before you even think about moving out, though. Once you got an income, look around at what you can afford. There's sites, newspaper ads, some papers even have their own websites, that you can check for private rooms for rent, or others looking for a new roommate, ect. Or see if you can get with a friend.
How old are you? If you're eighteen, maybe it's doable. Thirteen? Not so much unless you can go into care.
Anyway, assuming you're a legal adult, you should look for places that want roommates or are letting a room of a house instead of a place by yourself; it will probably be more cost effective that way. From there, you'll need a varying amount of money up front for a deposit and, of course, a steady job that will allow you to pay rent. If you have these things, start looking through local adverts and see if any are suitable.
Do you have money for a security deposit? Are you employed? You will need to show evidence of your income. Have you established any credit? Most places will not sign a lease unless you pass the credit check and they will expect you to pay for them to run it. Probably $20. Security deposits can vary from $100 to $1000 depending on where you live.