Don't be embarrassed. Everyone has a first day at the gym, and the smart people are those who ask questions.
You should have a membership person who tells you who the trainers are - most gyms provide some introductory instructions (free) to the new member. They should ask you what your goals are and help you with proper use of the equipment (you won't be using everything) and a class schedule. That's separate from the personal training you pay for. Most gyms give your one or two free sessions with a trainer, or at least have someone walk around with you to show you how to use things. They often have a large card they give you and they can write down the different pieces of equipment you select (together with their advice) and how much weight to use, how many repetitions to do, etc.) It's important that you start with less weight and not injure yourself, and it's important that you learn how to use proper posture to protect yourself.
It's not always good to observe other people - sometimes they do things wrong! The weights should not be clanging against each other, and it's important that you put things away when you are done. Put the weights back on racks - in order - and replace any stability balls or anything else you use. It's okay to take something off the rack and take it to an area where you are doing several things. I take a mat and a large stability ball, a medicine ball and a set of weights, and I alternate doing different things with them. I do something for my arms, for example, set the weights down, and do some abdominals on the stability ball, then I repeat this set allowing my muscle groups to relax in between. That's fine - just return the whole thing when you are done.
If a weight machine has a pin to change the amount of weight, you can leave it as is when you finish. If there are large weights that are manually loaded on and off, then you should probably take your weights off when you are done. There is spray cleaner and there is a paper towel dispenser to use when cleaning off the equipment - clean the seat and the back, and any handlebars that you grip. You don't have to scrub - just a wipe down. That gets your sweat off the machine for the next person. If you are doing a series of repetitions (such as 3 sets of 12 reps), then just clean at the end, not in between (unless you are getting up to use something else and someone is waiting for your machine). Then just spray the towel and wipe down, carrying the towel with you. If you use a treadmill or elliptical with heartrate sensors, spray the towel and then wipe down the equipment - don't spray the liquid directly on the sensors.
Buy a small combination lock so you can lock up your coat and pocketbook. I like to take a small hand towel in case I get really sweaty. I just drape it over the arm of the treadmill or put it on the floor next to the equipment I'm using. I also take water - gyms may sell bottled water but most people really prefer that you use your own reusable bottle rather than fill up their trash bins. Do not take food in there! If you need an energy bar, eat it in the lobby area (sometimes gyms have a small area for the stuff they sell, so you can eat your own thing there). In fact, I have a class that celebrates the members' birthdays, and the gym lets us bring a low-fat cake in there! But definitely not in the workout areas.
You can use an iPod and some gyms have TVs you can connect to by plugging into the system to get the sound. Ask for help.
A lot of people read on the treadmill or bike - sometimes the gyms have a big magazine rack so feel free to choose something. The equipment may have a built-in book rack or a separate rack you can hang over the control panel. You can bring your own stuff too - in fact, I just rip off the address sticker and recycle my own magazines there for others to enjoy.
If you plan to shower there, take some travel size materials and your own towel - I like a beach towel. You might want flip flops too. You can use their hairdryers and put your stuff on the counter while you are doing your thing, then clean up when you leave.
In bad weather, don't wear your regular sneakers to the gym. Use "indoor only" sneakers so you don't track rain, salt, sand and mud into the workout areas. Put your boots or "outdoor shoes" in the locker with your coat and purse.
A lot of people get very focused during their routines and don't chat, so don't be offended if people roam around from machine to machine without talking. The gym may have a posted policy asking you to keep chatter to a minimum, but that doesn't mean you can't ask a staff member for help or ask someone if they are done with the equipment you are waiting for. If there are extra machines, it's nice not to crowd somebody by taking the machine next to them. But if it's crowded, it's crowded and take anything that's open.
You might want to keep a beach towel or bath towel in your car to sit on and lean on so sweat doesn't get on the seats.
I'd recommend a sports bra with good support and a light sweatshirt or other cover-up for afterwards - you don't want to get cold. I like an open necked tee shirt or something made of wicking materials so the sweat doesn't collect on my skin.
Also, it's FINE to ask the staff what the customs and routines are at that particular gym. Phrase it as "What do most people do with....?" or "What do you suggest for.....?" That's what they're there for and that's what you're paying for. And ask what classes are good for someone at your fitness level and with your (lack of) experience. You want an instructor who welcomes new people, who helps them, and who doesn't assume you know what you are doing. Try not to hide at the back of the room where you will not be able to observe the instructor or hear what he/she is saying.
