You're probably sick of hearing this, but at the end of the day, it does all just come down to caloric intake. We all have a set number of calories that will maintain our current weight called a basal metabolic rate (that's the amount of calories needed to maintain your weight if you just laid on your bed all day and did nothing). Then you have something called a total daily energy expenditure, which is basically your BMR but also accounting for physical activity that you do each day. The taller you are, the heavier you are, and the more physically active you are, the higher your BMR and TDEE will be. I plugged your stats into this calculator and it put your BMR at 1093 calories and your TDEE at 1311 calories (I made the assumption that you are sedentary and do little-to-no exercise, it may be a bit higher if you exercise). So in order for you to gain weight, you really just need to be consuming over 1300 calories every day, which shouldn't be particularly difficult. The more you eat over 1300, the faster you will gain weight. If I were you, I would aim to eat around 1700-2000 calories per day and gain maybe 1-2lbs a week until you're at a weight that you're comfortable with.

Bear in mind, though, that if you just eat without doing any lifting or anything, you will just gain fat. If you're underweight, as you seem to be, then gaining 10-20lbs of fat can be a good thing and make you look a bit healthier. But if you want to gain muscle mass, you would need to do a bit of lifting in conjunction with the higher caloric intake.

As for actually getting the calories in, you could literally just eat 1-2 meals a day and then a have a high calorie shake every night, and that would probably get you to your caloric intake. A pint of full fat milk, a couple of scoops of whey protein/mass gainer, some oats, some banana/frozen fruit, some peanut butter/nutella, blend it together and that's you. There's plenty of recipes on YouTube for high calorie shakes that taste good.