Transcribing music is an incredibly educational process and is wonderful ear-training for aural development. Especially as a beginner, it is important to begin with a wide scope, and gradually narrow the focus to eliminate otherwise distracting information. Often, beginning musicians are overwhelmed with all of the aural information, and so the model I will outline below is one that I have used and taught that separates the aural experience into distinct layers. Some of these layers are not necessary or applicable for every piece of music, but are important to be aware of nonetheless.
These steps should be taken chronologically from top to bottom:
Determine which instruments are playing. It sounds simple, but if you want to transcribe a cello part for piano, it's helpful to know how they sound differently than, let's say, a the viola section.
How fast or slow is the music? Again, sounds simple, but if you think the music is twice as fast as it really is, your transcription will need to have a lot of editing.
If there is a tonal center or tonal region, this will help you determine which notes are more likely to be relevant than others. If you know the music is in the key of F, you'll also know it's unlikely you'll need to use notes like C# or G# unless something significant is happening. Similarly, if you know the piece uses a  trichord, you have something on which you can hang your hat.
Form / Phrasing
Understanding how the piece is organized (its underlying structure) greatly helps in transcribing music. This is especially true of “pop” music because once you transcribe each element of the song, the rest of the song is really just a matter of copy and paste. Understanding Sonata-Allegro or Rondo form can greatly help you create expectations of tonal or thematic relationships. The more you can expect, the less you are surprised by, and the quicker the transcription goes.
When possible, do your best to determine the rhythms of the notes you wish to transcribe. If the piece has a time signature or changes time signatures, make sure you understand what they are and when exactly they change. Knowing the tempo will also help you determine the time signature and which rhythmic durations represent the sounds you're hearing. If the piece is without signature or is improvised, then it is important that you know The Rules of the Piece. (see below) Rhythms can either be written exactly, or, if you're pressed for time (as many musicians are), it is very common to use dashes for longer notes and dots for shorter notes. Even making general graphical expressions can greatly reduce stress while transcribing.
Once the rhythms are notated or approximated, the next item is pitch. Often people try to transcribe both pitch and rhythm simultaneously, and this consistently leads to sub-quality transcriptions from the beginning musician. Over time, musicians develop the ability to notate both simultaneously, as certain patterns and frames of expectation become more developed. Using the information determined from the section in this answer about Key can greatly help here. Often, students should use familiar songs to help them master interval recognition (such as “Somewhere Over the Rainbow” for an octave). Students and beginning musicians should also learn to recognize common chord progressions, usually represented through roman numerals. An example of I-vi-IV-ii6/5-V-I is very common and found in tens of thousands of pieces of music.
Determine which parts are louder or softer than other parts. Determine when the music is growing or receding in volume. This knowledge helps you decide which parts are more or less important than others. It is also necessary for creating an exact musical transcription.
Much like dynamics, articulations illustrate how a given note should be played. Articulations help define the character or style of the music. To help other people recreate the style / character of the music you hear, it is important to know how the music is being articulated, and what symbols represent the appropriate articulation. (Imagine how silly Samuel Barber's Adagio for Strings would sound if everything was played staccato.)
Expressive indications can describe a lot about the character the composer wishes to convey. Whether it is “Allegro ma non troppo” (Quickly but not too much) or “Langsam” (Slowly) or whatever Mahler wants, having an idea of the original indication, or creating your own indication can help communicate the music with other people.
Whether you're transcribing a pop song, an art song, or a folk song that has been passed orally through generations, you should always do your best to notate and/or translate the text as accurately as possible.
The two below are not necessarily chronological steps, but are important in the transcription process.
Know the Rules of the Piece
This is especially important for new music. If you've got a specific recording of a John Zorn, Earl Brown, Pauline Olivieros, Ligeti, Messiaen, or a host of other contemporary composers you're interested in transcribing, it's really important you understand how the piece works. For indeterminate or aleatoric music, the music can often be represented abstractly through graphics instead of traditional musical notation. If using this strategy, it is even more important that the music and information is conveyed as clearly as possible. After all, another musician will not only be tasked with understanding the rules of the piece, but your transcription as well. There are some conventions for graphic notation, but that is a bit more specific and can be developed in a further discussion.
Use Historical / Theoretical Knowledge
As with many of the other layers, using historical and theoretical knowledge can save a lot of time in the transcription process. For example, if you're transcribing a Haydn string quartet, you can use historical knowledge to know that it's unlikely you'll find tone rows or hexachords. Therefore, you can rule out a lot of expectations of sounds. As I said earlier, reducing surprise makes transcription move more quickly.