Ignition key an lock the second one and does not have a chip

Hi guys, we have a question. We bought this vehicle from the auction. It did not have a remote and its key looks like it is the second one and does not have a chip. Couple days ago it started to live its own life. The alarm starts right when open the door, and cannot lock doors without alarm starting after unlocking. Then, doors started to open by themselves. I lock them and they are unlocking right away. So we think it is a security problem, is it? We are afraid to drive! We think to buy a remote, but this is a question, do we need to buy just remote or a new key, too? How does this work? We know that remote is self programmable, but would it work if we buy it separately? Or do we need to buy it with key and program both?