First off...don't use your calculator, it's ok for checking your answers but getting used to using your calculator for basic problems will sink you once you end up in a situation where you're not allowed to use it. That and, you shouldn't need a calculator for calculus if you actually know what you're doing.
Second, you say that you're good at doing operations but don't get the abstract concepts of calculus. Calculus is in no way abstract, it's completely logical. That being said, if the reasoning is difficult to understand - don't worry about it. Very rarely will a professor ask you WHY you do something, a good majority of professors only want to see that given a problem, you know what operation to perform on it, and that you can do that operation correctly. Memorize the rules and it's all good. Very few people actually understand all the nitty gritty of math and if you're not a math major, stressing yourself out trying to learn it is completely pointless. There's only so many problems they can give you to solve in calculus, so do practice problems and eventually you'll have seen pretty much anything they can throw at you.