Assuming the music was purchased by you and was not obtained from an outside copy, you will typically be in the clear when it comes to adding the music to your computer library. It is assumed that the music will be for personal use only. Therefore, you can also transfer the music from your computer to a personal music device.
People get into trouble when their intention for copied music is not just personal. Copying music to distribute it to friends or sell it on the Internet is illegal and can lead to harsh consequences. Additionally, copying music to your computer from unauthorized sites is also against copyright laws.
Copyright laws are designed to protect the creative works of musical artists. Provided your intentions for the music are non-commercial, you generally have no reason to be concerned. However, crackdowns on illegal music piracy are happening. If you ever have any questions regarding a copyright situation, you should research the law to ensure you are staying within your rights as a consumer.