You ask what your dad needs to do to make you the owner of his old house. The answer is he can simply give it to you. To do that,
he needs to prepare a deed transferring legal ownership of the house to you, and have that deed notarized. Then he must record that deed at Recorder's Office for the county where the house is located. Plus the state or County may have other forms that need to be filed when the deed is recorded.
If you father's interest in the house is worth more than $13,000, he would technically (and legally) be making a federal taxable gift of that property. In theory, he would be required to file a federal gift-tax return. No tax would be due, because, as of now, he could transfer property worth up to $5 million before tax must actually be paid.
A more serious concern would arise if the house had risen significantly in value since your father bought it. If that's true, federal rules on "stepped-up basis" mean it would be better , from a capital gains tax viewpoint, to inherit the house after your father's death, rather than receive it now while he's alive. The stepped-up basis rules are somewhat complicated, and this may not be of concern to you. It would not affect your father one way or the other. So I won't bother to explain the stepped-up basis" tax rules. You can always google that if you want to learn about it.