Well you should definitely take advantage of the free weights. Bent over dumbell rows, tricep extensions, goblet squats. Don't worry you will not get bulky by lifting weights. Ask anybody trying to gain muscle mass it's not easy. You can look on bodybuilding.com for free workout programs to aid in finding what works for you.
Many gyms have trainers who can help you get started without signing up for actual paid training.
If you're intimidated by free weights (many ppl are but believe me it's just all about picking things up & putting them down) you can very safely get started on the machines without much to learn.
Use Google though. Look up terms like 'simple beginner gym exercises'
'Beginner free weights'
'Weight training for beginner'
'beginner gym circuit'
'how much weight should I lift'
... Stuff like that.
You can gain a lot of benefit starting really simple with some major muscle groups (chest/bench press, pullups/lat Pulldown, squats, ...) & slowly work into more specific exercises.
As previously said by another, don't worry about 'bulking up' like a Hulk... Weight training is very healthy & will make you look great. Ppl, especially women, only get Hulky with a LOT of effort and very careful, educated planning to get exactly those results.
Whatever you do, you should definitely KEEP TRACK of what you do. Some gyms have a paper form you can keep there in your own file, I use an app called 'simple workout log'. You can make your own names for exercises, & keep track of weight & numbers. It's simple.
You start off not knowing how much to lift but if you keep track, very soon you find a good level that you find challenging enough. Then, you can refer to your history & be sure that you always try to match or push just a little harder next time.