Ok, so I'm not exactly a bodybuilder myself, but I have been pretty skinny all my life and despite years of (off-and-on) working out and trying to gain weight have finally started making gains this last year so maybe my experience will be useful. For a naturally skinny guy it really is mostly about eating enough food so that's the bulk of my advice.
1) Get a calorie counting app:
I use LoseIt! but there are many. Until I took a comprehensive survey of how many calories I was eating on a daily basis, I had no idea what my intake was, and because I'd never had to worry about calories before I really didn't have a good intuition for how many calories or grams of protein are in common foods. Use it for a few days without altering your diet and then use it as a guide to make sure that you're gaining enough at every meal and throughout the day. It will train you to feel a certain amount of fullness and you'll start to notice when you don't have food in your belly. After a while you probably won't need it anymore and (pardon the pun) rely on your gut. As a vegetarian, I'm sure you're used to combining grains and pulses, just make sure you're getting enough of it all and lay in on the higher-protein foods like egg, cheese, milk, whole-grains, peanut butter, and tempeh as well as good fats like flax, olive oil, and avocado.
2) Cook in bulk:
Never, ever, ever cook just enough of something for one meal. This is useful advice for anybody who doesn't want to be eating out all of the time, but it's especially useful for someone trying to gain weight. If you're on a meal plan in college USE IT. Having access of tons of ready-made food in an all-you-can-eat-ish scenario is definitely to your advantage.
3) Weight gaining shakes:
I'm a little sketched out by store bought weight gainers - many of them are full of sugars, empty calories, or just a long list of chemicals I don't want to sort through - so I just buy protein powder that has a simple, short ingredients list and get into the habit of making homemade shakes in the morning to have throughout the day. They taste way better and are certainly better for you. One that I make often is (per serving):
3/4 cup of milk
2 tbsp peanut butter
1 scoop protein powder
1/4 cup whole oats ground in a coffee mill
(1 tbsp of flax oil)
(1 tbsp probiotic)
4) Don't get overwhelmed:
Think of fitness and nutrition like learning a language. The amount of information (and misinformation) available to you when you're starting to work out can be daunting, but the important thing is to be moving in the right direction. You'll never learn a language if you try to wait until your grammar is perfect to try to speak, and similarly it's impossible to learn a new language without going through a period where you look/sound like an idiot. Just pick a workout plan that looks sensible and start lifting regularly - you can learn more as you go. Having said that, it's important that you don't injure yourself and maintain proper form so...
5) Make a gym buddy:
I always assumed that I would be too wimpy, uninformed, or unadvanced to be a good gym buddy to anybody until I got better at it, but that was pretty silly. If you're a fun enough person to hang out with for an hour that's all most people are really looking for, and many people enjoy being able to pay forward all of the information that other people gave them along the way. Find one at your gym, though other friends, or online, pick someone who knows more than you and has a reasonably comparable physiology/metabolism. Just take any non-trainer's advice with a grain of salt and err on the side of caution.
Give it time:
It's a slow process for everyone, and also consider that everyone's metabolism slows with age. The guys who bulk up easily when they're 19 are going to have to work harder to keep weight off when they get older whereas you'll be settling into a good range for the kind of body you want.