1. Study the song formSongs use repetition to create familiarity and variation to create interest. By writing down the song form (and maybe even more details - like an analysis of the chords and key areas or when a particular riff or melodic phrase appears and reappears), we can understand the song better.
2. Study the harmonyDoes the song stay in one key? Does it moves through many keys? Is it in a major key or minor key or both? Is it modal? Is it pretty static or does it change chords every two beats? What is similar between two chords that seem so different? By thinking about the harmony and figuring ways to voicelead and connect chords - the song will be more a part of you. Memorizing the chord changes is also very useful in further developing your personal approach to the song.
3. Study the melodySame thing like the way we looked at the harmony. In fact, we should probably study the melody and memorize it first before looking at the harmony! Also, it's useful to look at how the phrases are constructed. Are they long? Are they short? Do they have a call and response type of idea? Are the phrases balanced or not?
4. Study the rhythmsAre there repeating rhythmic ideas? Is is very heavy on the downbeat? Is there a lot of syncopation? Where are these syncopations?
These are some things we can think about when we learn a song. Also, I find that it is very useful to learn the song's bassline, melody and chords (accompaniment) separately to be able to hear the parts. This thinking is applicable for any style of music that has these elements. Some styles may just have one or two instead of all three - particularly non-western based traditions that rely more on drones and melodies.
Most of all, remember to have fun and enjoy the process!
Till later, play on!