Individually I get the with words car and big (which is being modified). But in between the words I get a "tune" sound. what is that?
if you are talking about the (-un) at the end of Sayyarah, then this is called Tanwin or Nunation. You can imagine it as the equivalent to A/AN in English with indefinite nouns. Since Car (سيارة) is a feminine word and ends in H in normal conditions, changing or adding to this word would change the H sound to T, and hence it is written with T-Marbuta (ة) and not a regular H (ه) nor a regular T (ت).
'un' in sayyaraTUN is grammatical. According to the position in the sentence. It could be tAN or tIN. But for the beginning lesson, just follow the instructions.
You could ommit in conversation. Sayyaratun jadiidatun = sayyaratun jadiidah = sayyarah jadiidah.
Ta marbutah (ة) will be silenced become 'h' not 't'.