Learning any language is a challenge, not just for the new vocabulary, but because you are also forced to learn an entirely new form of grammar in many cases. ASL is no different.
Find a teacher for the best results, or sign up for a class. If you cannot do this, you can find the necessary resources for free online.
Step2
Read and absorb as much information about the grammar as possible. If the reason that you wish to learn is to communicate with a deaf person, ask them for their help in correcting your attempts.
Step3
Practice your sign language often, both for the grammar aspects and the vocabulary.
Tips & Warnings
Most people teaching sign language tend to prefer to get as far away from English grammar as possible. They might say "teacher me" instead of "me teacher". Both are equally correct.
If you are simply communicating with others and not looking for certification, keep things simple by phrasing things closer to English. Example: "my car".
Do not expect to become proficient right away. Learning takes time and patience and lots of practice.