Practice reading musical rhythms aloud to grasp the tempo or timing of the music before attempting to sing the notes. Begin with simple rhythms in which the quarter note gets one beat and then progress to more complicated rhythms. Count the rhythm using numbers, one number for each beat.
Learn the "do" syllables for singing the pitches: do, re, mi, fa, so, la, ti, do. The simplest method requires you to sing "do" for every C, "re" for every D, "mi" for E, "fa" for F, "so" for G, "la" for A and "ti" for every B. Don't worry about distinguishing between flats and sharps in the beginning.
Match the pitch and rhythm by listening to short musical phrases and then imitating both the pitch and the rhythm, using the "do" syllables. Memorize both the pitch and the duration or timing of the pitch.
Increase the length of musical phrases and the complexity of the rhythms. Listen and then repeat again and again until you gain confidence in your ability to match the phrases from memory.
Consider finding a voice coach or another singer who can judge how closely you're matching the pitch when you repeat what you're hearing. Some ear training software records and displays the actual notes you sing so you can immediately see whether you've accurately recreated the notes you heard.