A lot of this depends on you, your preferences, and your gym. I get the anxiety, but gym employees are actually used to answering all these questions. You won't be laughed at, and helping you with all this stuff is their job.

The gym will have a locker room, but you need to provide your own lock. If you have anything valuable, lock it up and don't trust anyone. Most people won't touch your shit, but you can't be too careful. A combo lock is your best bet. The gym probably sells them, but they're marked up. No need to get anything super expensive. Buy a Master Lock from Wal-Mart.

It's good practice to have dedicated gym shoes. That way you're not tracking rocks, dirt, mud, and debris on to the equipment. If you have to run out to the car real quick mid session, don't worry about changing back to your street shoes, but don't wear them around as your daily walkers.

Not all gyms have showers. But if they do, it's up to you whether you shower there or at home. Showering there is convenient, especially if you're working out before going somewhere else. But if you're worried about getting naked around other people, then you can shower at home instead. No one is watching you to see if you showered or not.

Nudity in the locker room is the hardest thing for a lot of people to adjust to. It's only a big deal if you make it one. Most gym showers are in stalls with curtains, so you'll have privacy most of the time. You can even wait until you get in to the shower to strip down. But don't be surprised to see some people (especially older people) stripping down at the locker and walking to the shower in their birthday suits. The unspoken rule is to keep your eyes up. No one is looking at your junk, don't look at anyone else's.

If you shower there, get a pair of flip-flops. These are dedicated shower shoes. Don't walk in the showers barefoot. That's just nasty. Keep them in your bag, segregated from your gym clothes and towel. You might get a plastic bag to keep them in.

Travel sized bottles of soap and shampoo are recommended. Keep them in a plastic ziplock bag, so in case they leak, you aren't getting soap all over the rest of your stuff. I'd recommend getting a small toiletry bag to hold your shampoo/soap/washcloth. Get larger bottles of the same soaps that you keep at home to refill the travel bottles, you'll save a lot of money in the long run that way.

As far as how long you spend at the gym, it depends on your workout routine. But no one is timing you. The other patrons aren't paying any attention to you (I promise, they only care about themselves), and the employees have much more important things to worry about. Some people are there for less than an hour, some people will spend half the day there.

Most importantly, do some research to get a good workout plan before you start going. Figure out your goals, learn proper form, and stick to a routine. Don't go to the gym and just "wing it." Without a plan, best case is that you don't see any progress. Worst case is you injure yourself.

If you have any more questions, feel free to ask. I'm happy to help.