What it sounds like is you are suffering from tension and poor hand and arm position. Hopefully you have a piano teacher as they can help you. You are bit too early on the piano technique exercises. After you've advanced a bit more, there are some books you can purchase that have a multitude of exercises in them. Recommending them now is a bit too early as they can cause muscle and tendon damage if they are done improperly.
First ensure you are sitting properly at your piano/keyboard and it is the proper height from the floor. You don't want something that is up too high, such as on a table, as that can cause fatigue in your arms, neck and back as you reach up to press the keys. You also don't want something that's too low as that can hurt your back and neck as you lean forward.
So, if you don't have one, get yourself a keyboard stand, unless you have a digital piano that is on a stand.
You want to sit up straight with your feet flat on the floor. Don't worry about pedals at this point (if you don't have any, or even if you have any), as they are not important to you in this early stage of the game.
When you sit at the piano, your arms should be slightly higher than the keyboard so your fingers can press the keys comfortably. Your hands should turn in just slightly so all the fingers, including the thumb and the small fingers can play in the center of the white keys (on the big part in the front).
When pressing the keys, curve your fingers slightly like you are holding an orange or a small ball. This is a natural position for your fingers. Playing with flat fingers will get in your way later, and will cause a sloppy sound. You also want to play to the bottom of the keybed and evenly so that the tone is firm, concise, and clear. This will allow you to control the dynamics evenly when it comes time for that.
Remember the whole key (pun intended) is for a relaxed arm, hand, and neck, and to play evenly. Once you get this down, you can then worry about finger independence exercises, which at this point in the game can actually hurt you physically.