(This answer has been moved from the comment section and reworded a little bit.)
I would recommend going to a Saturday morning service over a Friday evening service for a first time experience. Friday evening is very sweet but short, and you won't really be able to get a good sense of what a service is all about. Plus, you get to see the Torah Reading Service on Saturday. Expect to spend a good 3 hours in synagogue Saturday morning.
Women and men sit in different sections of the synagogue. Usually (but not always) men and women enter the building through the same door, but then there will be separate doors on the inside which lead to their respective seating sections. You might need to look around first to see which door other men and women are entering before you go in.
In terms of what to do and what not to do: Women should wear a dress or skirt that comes down to at least a little below your knees, and not wear slacks. Men should wear a suit or at least a dress shirt and pants. Your top should cover your arms down to at least a little below your elbows. Do not wear a scoop-neck top. Modest jewellery is fine. All men wear a head covering, but only married women wear a head covering. If you're an unmarried woman, do not wear a head covering. If you're a woman, you do not wear a tallit (prayer shawl), either. Do not bring a purse or pocketbook in to the synagogue, and do not bring in cameras or cell phones, either. Really, don't bring anything in with you.
Sit when everyone else sits, and stand when everyone else stands.
Yes, you can certainly hold the siddur and the humash (Bible). As Shalom pointed out, however, be very careful not to bring any books into the restroom; leave them on your seat if you need to go.
There are certain points during the service when we are not allowed to interrupt the prayers by speaking. If you ask someone a question and they hold up a finger to signal "Wait a minute," just wait a minute and they will answer you as soon as they are permitted to talk again. Don't worry about it; it happens all the time that people interrupt other people at a point when they cannot respond. It's completely normal and usual.
There will always be someone who will be happy and honored to help you follow along in the siddur. Let the details wash over you; if you decide to pursue it, you will learn the details in time. Know, too, that you will not be the first or only person coming in who has never seen a service before. Everyone is at a different level of learning and experience, and it's all okay. May you receive blessings on your journey.