Is the God of the Old Testament the same God now?

Have you ever heard another say "the New Testament is about a God of Grace and Mercy, the Old Testament God was a God of wrath!" How much of the Old Testament God is the same as the New Testament God? Do we live today according to different rules, a different covenant than the Old Testament covenant? What about Old Testament laws, should we stone people for committing adultery and teenagers for rebellion? Why do some preachers teach the keeping of Old Testament laws such as tithing and keeping the Sabbath? Why do different preachers pick and choose which Old Testament scriptures they will apply to our lives today, laws we should obey? Should we not eat pork and if you are a farmer, should you let your land rest every seventh year? Are we truly serving the same God of the Old Testament or did God change somehow when he offered his only begotten Son for the sins of the World? What is the New Covenant? Looking forward to it!