I do not know any authentic source (Quran or Sunnah) which explicitly gives in details the contents of the Tablets/Tauraat given to Musa (Peace be upon him). Nor is there proof that the contents were the ten Commandments. There is a Tafseer, that I follow, and am going to use to try answer your question.
All messengers had nine/ten commandments/Wasayah which was the base of the deen, and they gave to their people. These nine/ten commandments are mentioned in Surat Al An'aam Ayah 151-153
and the ten Wasayah in those Ayat were the ones which were given to the previous peoples on the tongues of their messengers
ذَٰلِكُمْ وَصَّاكُم بِهِ
This He has commanded you
And so this is what Musa gave, and then after that the Tauraat came to explain it and expand on them, just like the Madani Suar came and expounded on that which came before in Makkah.
Sources: تفسير المنار