Your gym should offer some sore of free orientation program. With this, depending upon the size of the gym, you make an appointment and someone will take you around the gym and introduce you to different equipment. If it's a large gym, this may involve more than one appointment as you are introduced to different parts of the gym and different equipment at each one.
Until the appointment time comes, there should be someone on staff to show you how a piece of cardio equipment works. Trust me, they really don't want people to use the equipment improperly.
As far as group classes go, like Spinning, Pilates, etc., there is usually a printed explanation you can get as general explanation of each class, with a list of instructors, and requirements for attendance. Normally, there are days and times the instructors show up a little early, especially Spinning, where they are wiling to give you a more detailed description and help you get set up on your bike/equipment. There should be a representative at the gym to let you know how to get set up for any group class and know the instructors schedules and basic routines.
You don't have to have a personal training session to learn about the equipment. In a responsible, reputable gym, they keep people on staff to answer questions, help patrons, and look for safety issues. They want you to be safe and to use proper form.
It is also important to recognize that the personal trainers are often paid by their personal training appointments rather than by the hour. They may or may not be separate from the other staff that takes care of equipment and general assistance of patrons. Don't expect them to design a whole program for you if you are not their "client." Personal training sessions are different from orientation appointments. Orientation generally gets you a tour with an explanation of how to use equipment (but make sure to ask) and personal training appointments involve designing a program specifically for your fitness goals. The pt appointments are usually associated with an additional fee whereas your orientation is generally a part of your gym package.
There is so much available at a full-service gym. Definitely take the tours/orientations to learn all you can to see what you like.
Sorry the post is so long.
Hope this is helpful