Well the first thing is, you wont get bulky form lifting weights.... not even by lifting heavy weights unless you do it like 5-6 times a week, keep increasing weights over a 1-5 year period, but even then you might still not be bulky.
Some may see me as 'bigger' but im not bulky and ive been strength training since 2013 roughly, though of course as i love cardio that has been a part of my workouts as well... but you dont need to worry about getting bulky or bigger because you lift weights. Muscles dont grow fast, its not like you do 2 workouts and suddenly your biceps are alot bigger. It takes time and hard work.
And if you want to get stronger than focus on using weights where you can do 12-15 reps.
I would recommend you start by asking someone at your gym if they can give you some advice or tips, then also getting some exercise/nutrition books can be a good idea and then using this site: http://www.bodybuilding.com/exercises/ can help to show you how to do the exercises properly and such.
I dont personally recommend at home training, it can be very effective but i think its better to go somewhere else to workout. Though for some they dont have the money or time to go somewhere else to workout, but for me as i have spent so many years secretly exercising in my room, now my home and room is a place where its relaxing and i wouldnt even have the motivation to exercise at home. But if you feel it works for you, its not an obsession or a 'must exercise, must compensate' and you feel you cant rest at home, then why not. But i think the gym is great... at first it can be scary to go into the weight lifting room, feeling like a newbie and not really knowing what to do. But Fake it till you make it. Thats the best advice.... do some research, make yourself a workout plan, put on some bad ass musis, focus on you and your workout and forget the people around you because the fact is, they are all so focused on themselves and their workout and if someone is staring at you well then they arent there to workout and you should still just focus on you and know that you are working hard!
But start off with basic exercises, find your strength, find your enjoyment... if you dont enjoy lifting weights well there are lots of other forms of exercise which are great. So you dont have to lift weights if you dont enjoy it.
Asking questions is good to do as well :)
Also this post: http://www.nerdfitness.com/blog/2014/01/14/strength-training-101-where-do-i-start/ can be helpful :)