Settle into the rhythm of the music and identify the beat. You don't need to count off any numbers yet, but find the regular pulse of the song and mark it by tapping your foot or snapping your fingers. Continue until you can do this without even thinking about it.
Find the "one." This represents the start of each new bar of the music. It is usually marked quite clearly by the beginning of the melody, or the start of the lyrics. Alternately, if the lyrics or melody start a little later in the bar, the end of each phrase will usually be signified with a drum fill or a crash. Look for repetitions in the phrasing or melody, and try to pinpoint where each phrase begins.
Start counting on the "one," and keep up with the rhythm, counting each beat off as you were before. Keep going until the end of the phrase, then start again from one when the next phrase begins. For instance, the nursery rhyme "Frère Jacques" has four beats to each bar, with one beat for each syllable in the words "Frère Jacques." Try singing the melody but replacing the words with "One, two, three, four."