I recommend push ups: Figure out the max amount of push ups you can do at one time. Do that amount 2 times a day (so if its 20, you would do two sets of 20 which would be 40 total in one day). I recommend doing this 4 times a week. Monday, Tuesday, Thursday, and Friday. If you miss a day, use the other days as make-ups, but be carefull, those days are there so that you can rest your muscles. After each week of push ups, increase your max by 1. So if it was 20 push ups a set last week, do 21 this week, and 22 next week, etc. You can do this with sit ups with some modifications. Hope this helps!
I also will say you have to do push ups and eat good food. I usually do 3 set of 25 push ups, Push ups are good, because it will not harm your muscles. After push ups join gym but don't lift heavy weights, it will harm your muscles.
Take this from someone who has actually been in your shoes I used to be scrawny but now becoming a bodybuilder. I would recommend you to eat an extra 500 calories and get a gym membership so you can do exersices and don't have to worry about being alone or buying more weight. After you have all that exercises like bench press, dips, pull ups, curls, squats, skullkrushers, etc. will help you gain muscle not only on your arms but the rest of your body so start now and enjoy on becoming a beast.