Everyone navigates a public restroom differently. Some just go into the stall and do it; others have to wipe the seat with germ killer. Here are some tips that I have learned from my own experience. (Please note that these are for women's restrooms, not men's. Sorry, guys, but you're on your own there.)
Don't crowd your stall-mates.
As it is with men and urinals, it is with women and stalls. If you walk into a public restroom and are the first one there, don't be a jerk and take the middle stall. By taking a stall on the side, you allow the next person to take a stall that is not directly next to you. And if there is someone already in the stall when you enter the restroom, do not sit down right next to them. That's just awkward. Unless it's crowded and that is your only option, give them their space and leave an empty stall between the two of you. When it's crowded, like at a concert or a fair, it's alright to sit right next to someone else because there are a lot of people in the room and generally the people in line are talking with one another. But when it's just the two of you and you're side by side in a quiet restroom, it may make someone uncomfortable.
Check for toilet paper.
It is generally a good idea to check the dispensers before sitting down. I know it's hard to remember to do so, but even if you can train yourself to glance at the dispenser as you sit down, you'll have a chance of moving to another stall. When I go to the bathroom, I generally take some of the toilet paper and give the seats a quick rub down. Not only does this remove any drops from the toilet seat, but it also confirms that there is indeed toilet paper available in the stall.
If you forget to check...
We've all done it--sat down, done our business, and then realized the stall is completely out of toilet paper. When this happens, you have a few options. If you're lucky enough to have another person in restroom, you can politely ask them to hand you some T.P. under the stall door. Don't be shy about asking--generally, people are happy to help a sister in need. If you're alone, you can either wait for your savior-to-be to enter the restroom and then ask them for assistance, or you can use a toilet seat cover. No, it is not necessarily a pleasant experience, but it gets the job done and gets you out of there and back to whatever you were doing. The toilet seat covers are usually found above the toilet seat and with a twist of your body you can pull one down.
If you happen to be in a stall that lacks toilet seat covers and there is no help to be found, you're going to have to get creative. It might be a good idea to carry a small package of tissues in your purse/backpack/fanny pack/what-have-you for just such an emergency.
I know that many people use toilet seat covers, but I've never cared for them. Often times they fall in before I even sit down, and then they won't go down when I flush. Furthermore, I find them uncomfortable and feel as though I am going to slip off the seat at any moment. Another problem I find with seat covers is that they don't prevent my rear from touching drops on the seat, whether they be from the toilet violently flushing or something less pleasant. My alternative fix is that mentioned above: take some of the toilet paper and rub the seats down. This is a sure way to remove any wetness, and it also forces you to check for toilet paper before sitting down. Two birds with one stone, so to speak.
First come, first serve.
Many people are uncomfortable doing "number two" in a public place, but in some cases (such as in college dorms, on vacations, or at a rest stop) this is inevitable. Should you enter a restroom needing to go "number two" and someone is already there, sitting rather quietly, assume that they are in the same fix as you. So now you have two options: you can both sit there in silence waiting for the other person to leave, or you can be mature and courteous and try again later. Generally speaking, whoever's there first has the right-a-way. Of course, if you're one of those people who just don't care, you can politely take a stall as far away from the other person as possible and quickly go about your business, and then leave them to theirs.
There is a time and place for polite conversation.
This is not it. Please do not try to strike up a conversation with the stranger in the stall next to you. Under any circumstances. Just don't. Please. Even if you begin a conversation while waiting in line, do not continue it into the stall. It's one thing if it's a friend or family member and you're sitting right next to each other, discussing something relevant. (When mothers and daughters use the bathroom before shopping, they often discuss their plan of action, such as which section they'll be going to first, etc. This is normal. Asking a stranger about their pet turtle is not.)
When you're done...
Make sure you flush the toilet!!! I cannot stress this enough! So many times I have walked into a public restroom to find every single toilet filled with nastiness from the previous visitors! Remember that not every toilet is automatic! Even if it is an automatic toilet, make sure it does flush when you're done! Please!
Once you exit the stall, be sure to wash your hands. I know some people cheat and don't use soap, or they don't use it because it's anti-bacterial or something. Personally, I've never understood it, but don't get me wrong. I'm not hating. If there is another person in the bathroom with you, make sure you wash your hands. You don't want people spreading rumors about you and your lack of hygiene in the restroom.
When you've finished washing your hands, dry them with either the paper towels or air dryer provided. If there is another person in the restroom, do not stand in front of the towel dispenser. This will lead to them either standing and glaring at you until you move, or reaching across you to get a towel, which can get awkward if it's a perfect stranger doing so. After you get your towels, take a few steps away from the dispenser to allow others to access it. Of course, if it's just you in the restroom, you can do whatever you want.
If there are no towels and you loathe air dryers...
Do not use toilet paper. Bad things happen. Although it dries your hands, it often comes apart and sticks to you in tiny pieces. Rather, use some of the toilet seat covers. They don't stick to you quite as much and it's easier to remove the pieces if they do. (Toilet seat covers are turning out to be handy little things, aren't they? I've even had friends who have used them as emergency blotting paper between classes. I can't vouch for this, never having tried it myself, but they seemed quite pleased with it.)
And when you leave the restroom...
It is considered a good idea to use a paper towel (or a bit of toilet paper, if no towels are to be found) to turn off the faucet and then open the door. This method keeps you from picking up all the nastiness you just washed off, and protects you from the filth of those who didn't wash their hands before leaving. Many people don't practice this (myself included, unless I'm in a particularly nasty restroom), but there is no harm in doing so. Just make sure you can access a trashcan after opening the door, so you're not stuck walking around with the paper towel/toilet paper all day.
And in conclusion...
Of course, everyone has their own method of navigating a public restroom. You might have read this entire article and deemed it a waste of time and myself a disgusting idiot with no idea of hygiene, and that's perfectly fine. We all have our own strategies. I hope the rest of you might have learned a helpful thing or two, and I wish you luck on your next public restroom endeavor.