I think it is a foolish idea to try and start eating more before you start at the gym. Resistance exercise should increase your appetite and muscle mass. You will likely need to consume around 2200 calories or more to achieve your goal weight. I am assuming you are around 5' 10" (around 177 CM). I would also suggest about 240 grams of protein per day. You are going to need to take a supplement. Whey protein isolate is low in fat and does not contain lactose. It should be ok for you to take considering your digestive issues.
As a second point, if you are not against eating meat for ethical reasons, I would suggest that you consider trying the Paleo Diet as Dave Liepmann suggested. I cannot offer you factual scientific evidence to support this as no studies have been done, but I have known several people who discovered that their digestive issues were in fact linked to gluten sensitivity and not to eating heavy or fatty foods. They tried Paleo Diet for 30 days and it completely changed them. I would be interested in knowing if Dave has anything concrete, rather than anecdote, he can add to this as I certainly do not...
For someone your age, I think it might be worth trying as it seems to be a good way to build some lean muscle in conjunction with a resistance training program. Otherwise you are just going to have to increase your protein intake to around 240 grams and ensure your are eating over 2200 calories per day in conjunction with a resistance training program.
Regarding your edits:
If you want to gain the weight, you need to eat the calories. My estimate was within the proper range for your height and weight. Ensure that you are consuming at least 2200 calories per day (and of that 240 grams of protein) and that you are doing so relatively evenly through the day. You may need to eat more and you may need to increase that as you get closer to your goal. You need to practice a resistance routine in conjunction with this or the weight you gain will be mostly fat. It is a biological fact regardless of how you feel or your previous experience. If your body does not have a need to store the additional weight as metabolically expensive muscle, it will not do so. The only other alternative is fat. If you want to start going to the gym as a "skinny fat" individual rather than just a skinny individual, that is your choice. Notice I am not saying weights, I am saying resistance. it could be a combination of pushups at different angles and hand positions, pullups, dips, and unweighted squats and lunges. But again, if your body has no need to store it as muscle due to your activity, you will just get fat. QED.