If 
in fact Moshe [Moses] used the sun, moon and stars as given for us to use in Genesis to determine the days, months and years, how exactly did they determine the specifically commanded times for our new moons, Shabbat and moedim 40 years in the desert where no crops grew, and being hundreds of miles from Jerusalem?
The new moons and Sabbath are not determined by crops, but by the moon and sun. As for determining the start of the yearly Feasts, Israel went by the barley crop in the Abib stage of development to establish the first month of the year. Abib is even the name of the first month of the Biblical year, Deuteronomy 16:1. Barley is native to the Mideast, and in the wilderness it doubtless grew wild back then as it does today. Abib barley aside, being in close contact with Yahweh as he was, Moses would need only to ask Him when the first month was.
Read “The Biblical Calendar” for an in-depth look at Abib calculation.
