How many times do we need to pray a day according to the Tora?

I have read in the Psalms that King David prayed 3 times a day? Is there written something about the timings of prayer in the Tanakh? I am a new reader, and it's a lot..

There is no requirement for prayer in the Tanakh.  Each person is free to pray as he chooses.

Ah okey, so it would be wise to follow the way of King David, i have read that he prayed 3 times a day.

But there is a requirement to pray 5 times a day in Islam. Will that apply to us, or is it irrelevant since we aren't actually converting to Islam?

This doesn't apply to non-muslims.  And even in the Quran there is no basis for 5 daily prayers.