I think that everyone who accepts the New Testament accepts that with Jesus came a different way of thinking. The Mosaic Law was from the Old Testament and included a lot of rules about how the Jews were supposed to live. The Pharisees were the religious leaders of that time, and over the centuries expanded this list much more than was written in the Bible we see. By the time Jesus came, he became disgusted with this set of rules, and tried to tell the people they did not have to be so strict. Many of those rules were about the Sabbath and what constituted Work. The Pharisees followed Jesus around and tried to catch him doing something wrong on the sabbath. When he healed people on the sabbath, for example, they got very upset and called it work.
The Old Testament applied to a different time. After Jesus came and died for us, people no longer had to make sacrifices at the temple. Jesus sacrifice was once for all time, all the forgiveness we would need, if we just believe in him. Many of the rules they used to follow took early Christians a long time to let go, including that it was ok to eat pork. people who were not born Jewish were becoming Christians, and these people ate other kinds of meats, and since no one needed to be under the old Hebrew rules anymore, they did not have to stop eating them. You should read in your Bible starting at Matthew, and if you are very interested in this particular subject, skip to Acts.
So yes, Christians accept both old and new testaments, but we realize there have been many changes since the old, including a relationship with God based not on works and if we are following the rules carefully enough, but thankfulness for his love to us. Check out some other Christians besides the witness friends you have, because witnesses are nice people, but how they interpret the Bible is more focused on the Old Testament based on rule following and their love for him is often genuine but a little messed up and confused by men telling them what to believe and how exactly to do everything they do.
I'm really sorry about the loss of your grandfather. Grandfathers are wonderful. Sure wish I still had mine around.