You discover your Strengths and Weaknesses by performing actions, tasks, processes, and procedures, and then evaluating your performance. Oftentimes, the evaluation must wait until you've engaged in an activity for a period of time, to get a true idea of your capabilities. Afterwards, you can make an informed decision on your performance based on a host of data.
Another way to discover your Strengths and Weaknesses is to ask the opinion of others. Ask a family member, relative, friend, a sports coach, a mentor, your boss at work, for feedback on a particular task you regularly perform.
Children are very honest in their innocence and will tell it like it is; they're not afraid to tell people in a roundabout way of their Strengths and Weaknesses. I'm reminded of a sitcom I saw a few years back. An uncle, who was a boxer, invited his nephew to his boxing match. The nephew was very excited and said, 'Great, and after the fight can I ride with you in the ambulance to the hospital?"
The uncle learned very quickly his Strengths and Weaknesses as applies to boxing.
Finally, professional, accredited tests and grading in a particular discipline will give you a well-documented record of your Strengths and Weaknesses in the discipline of your choosing.
You can improve your Strengths by continually applying yourself to the task; researching better ways to perform a task; getting advice from professionals in the discipline, and taking an academic course that will help you develop your skills.