Whenever possible, get inside the head of the writer or the composer of a piece you're working on. How did this song come into the world? Learn it by retracing the steps the artist took to create it in the first place.
Strumming the guitar Try to get it by feel first - I'm sure Bo Diddley never sat down and wrote out a strumming pattern in his life.
Learning a classical piece Imagine Haydn at the piano, transcribing the notes onto the page in small groups. Learn it the same way.
Mastering a specific guitar solo Find the scale shape that makes sense in that key, and play around with it until you find a few licks that match what you're hearing on the recording. You know that Hendrix was also just poking around until he found something he liked.
Figuring out an acoustic pop song Don't forget to use a capo if you've got a lot of chords that aren't available in an open position. Why would the songwriter be playing a bunch of barre chords on an acoustic? If it sounds easy, it should feel easy.
Learning fingerpicking on the guitar, or an intricate repetitive figure on any instrument Except for classical music, don't worry about making it identical to the recording - just get the gist of it. If the person who played the part on the record was being loose and spontaneous, you should be, too. Forget about the twenty pages of minute variations and just play.
Combining sections of a pop song Put the record on really loud and play along with it. Don't ask me "how many strums until..." Listen to the record, play along with the record, listen to the record, play along with the record, ad nauseum. The person who wrote it had to play it repeatedly to get the right feel.
Finding the groove of a song on the piano Tap it out on your legs, as though drumming, to get the relationship between the hands. Rock piano is very percussive.
Assimilating a rapid passage Play a few notes at a time in short bursts. You can't play the whole thing that fast yet, but right from the start you're teaching your fingers how it's going to feel when you do.
While you might not be able to follow this path all the way to the end, your playing will be stylistically appropriate even in the early stages of study on a given piece. In other words, it will sound like you understand the music.