"The" (definite article) is used when referring to something specific, whereas "a" (indefinite article) is used to talk about just any particular thing. In this case "yeni şarkı" is referring to a specific new song and "yeni bir şarkı" would refer broadly to any new song.
Beğenmek is generally only used with non-human objects an experiences; you wouldn't use it to describe a person unless you were deliberately objectifying them (e.g. in a sexual context). Sevmek is also more akin to "love" whereas beğenmek just means "like/enjoy".