If you are really weak right now, though, don't try and lift too heavy! Not without the guidance of a trainer. Get some free weights in the 10-15-20 pound range. If you have little upper body strength, this will be plenty of weight to get you started. Look for exercises that built up the delts and biceps online. Do sets of 10-12 reps with a one minute break in between. By the last rep you should barely be able to finish it--however it should NOT be all that hard when you first start. If you start with a weight you can barely lift for the first rep then you won't make it through your sets. When you are first starting out and are super weak, it's better to build up with semi-heavy weights and higher reps than super heavy weights and fewer reps--as long as you are still feeling the strain. This will allow you to get the form down with less risk of injury as well. So when people say lift heavy, this doesn't necessarily mean to go nuts! Just that 5 pound hand weights won't do crap for you.
You definitely need to eat a surplus of calories. MyFitnessPal is an awesome free app for regulating that. Also, sit-ups aren't going to do a thing for your shoulders and arms. Definitely go with push ups. You mentioned that they are harder--everything you do to bulk up should be really hard. If it's not really hard then it's not going to gain you muscle. Lose you some calories, maybe, but not bulk you up. In other words you are putting in a lot of effort for nothing when you do a bunch of sit-ups (especially since you don't want to lose any weight and want bigger arms and shoulders).