The traditional site of Mount Sinai, where God gave the Ten Commandments to Moses, is the subject of much debate and contention. Currently, the accepted site is the mountain named Sinai in Egypt; there are other supposed sites which are being investigated.
Goshen, or Rameses, is the name of the land settled by the Hebrews, who occupied it until the time of the Exodus. It is generally accepted to be in the Eastern Delta of Egypt, with good farming and grazing land for crops and cattle.
Using these modern locations, the best guess as to distance comes from calculating the distance between the Delta of the Nile River and Mount Sinai. Exact latitude and longitude are difficult, but if we calculate 30°54'N/31°7'E for Goshen and 28°32'N/33°58'E for Mount Sinai, we can calculate about 380km between the two points, or 236.12 miles. Naturally, the Hebrews would not have traveled a straight line during the exodus, so their distance would probably be greater as they detoured around mountains, valleys, and hostile nomadic tribes.