Playing in front of an audience is very different then when you are sat practising by yourself. Confidence usually wavers and it can be difficult to concentrate. However playing for family is less daunting than playing for the general public and they will definitely be more forgiving.
Look at the time signature and then count out loud as you play. This will help you understand if you are playing too fast or too slow. Once you have mastered the tempo perfectly, you can stop counting.
Practice sight reading and allow yourself to make mistakes. Scanning a piece of music gives you an insight into how it might sound long before you play it. Done well, this will make learning a new piece a much easier task.
You should celebrate any mistakes you make, not beat yourself up about it. Making mistakes is all part of the learning process. Make the mistake, be mindful about it and then make a particular effort to avoid it next time.
Ask For Help
Somebody who plays at a more advanced level then yourself did not get there without practice and help, so let them give something back to you. Hearing a more experienced pianist play the piece you are practising, can be particularly helpful. You will hear how it should sound and can apply that in your next practice session.