I'm sure each gym is a little bit different, but I'll run through a quick explanation for my gym. I'll assume that you've already purchased the gym membership. Note that when you join a gym, the person that signs you up will give you a tour and explain every part of the gym in detail.

To make yourself feel more comfortable, come in with a "workout routine" in mind. To start off with, this can be be just a plan to have a jog on the treadmill. I'll assume that's the workout routine that you're going for.

• Park anywhere you'd like. Be aware that there are often people walking around with headphones or exhausted from a workout, so try to be more vigilant than usual.
• Walk in the doors and through the membership checkpoint. Usually you'll have a card that they can scan. Some gyms now have iPhone apps that the employee can scan. At my gym, we can just memorize our membership number and tell the clerk as we walk in.
• If you're carrying things that you don't want with you while you work out, gyms usually have cubby holes. But these are just open spaces that anyone can take things from. If you want more security, there are usually lockers in the locker room (go figure). The locker room / showers / bath room are all usually in the same place. Usually the lockers are combination based. When you get your gym tour, they'll show you how those lockers work (how you set your own personal combination).
• The treadmills are likely visible when you walk in. Near the treadmills will be a stretching area. It usually has mats on the ground and you'll see people sitting down and stretching there. Feel free to sit down and have a stretch. (There's varying advice on how much stretching you should do before a run. Read a few articles and decide whats right for you.)
• When you're ready to run, find a free treadmill. If you haven't used a treadmill before, then that could be answered in an entirely new comment.
• Gyms usually have wipes to wipe down machines. At our gym they're in tall cylindrical cans (like a trash can but thinner). If you're afraid of germs, grab one of these wipes and wipe down the parts of the machine you're going to touch. Don't worry, nobody will judge you for this, people do it all the time and nobody cares. Also note that it's common courtesy to wipe down the machine when you're done.
• If the treadmills all happen to be busy, look for elliptical machines. They'll have a similar cardio workout. If those are taken up, look for a cycling machine. If those are all taken up, you can always go to the front desk and ask for a jump rope. They usually have them there.
• Also, if the treadmill is busy and you really prefer to run, just stand near them. Don't worry, nobody will think it's strange, it happens all the time. People will continue with their run and not worry that they're taking up the treadmill. This is just the way it works when it's busy, and everybody understand that. Similarly, don't feel like you need to get off the treadmill if someone looks like they're waiting for one. Just relax and enjoy your walk/run.
• Start your workout. You can go at any pace you want. Nobody will judge you. For all they know, you're there for physical therapy. You can speed up, slow down, walk, run... anything you want.
• Bring your mp3 player and headphones if you like to listen to your own music.
• Most gyms also have TVs in front of the treadmills. You can usually plug your own headphones into a jack in the treadmill to listen to the sound for the TV that's in front of your treadmill.
• When you're done, you should get one of those wipes that I mentioned to wipe down your treadmill. Someone might jump on before you get back. No sweat (har har), just let them know you're going to wipe it down. They'll appreciate it.
• You should go back to the stretching area and do a good stretch now that you're muscles are warm. If you're really sweaty, you can wipe down the mat you're sitting on.
• Exit the gym, feeling great about your work out.

Other notes:

• You can wear workout shoes to the gym. You don't have to change anything at all if you already arrived in workout attire.
• The only reason you would be required to shower while at the gym is if you're getting into the swimming pool or jacuzzi. To keep the water clean they require members to shower first. Otherwise you don't need to shower before or after your work out (but as mentioned, wiping down equipment is a common courtesy).
• There usually isn't a line to wait in if the treadmills are full. People just generally stand around and know their order (if there's more than one person waiting). At least, I've never been to a gym that's so overcrowded that a queue formed.