What happens when I have joint aches on both of my hands? Am I getting arthritis?

Arthritis. Yes this could be early arthritis. You can try taking some over the counter medication first. If it does not improve you need to see a doctor and get x-rays.
Maybe or maybe not. Joint aches can occur for a variety of reason. Arthritis is one reason but there are many different types of arthritis. This needs to be evaluated to clarify the reason. The hands have muscles, tendons, ligaments and each of these can ache. Circulation changes can cause aches. Nerve involvement can cause aches. We need clarification of what you have to answer your question.