The final 1040 reports income up until the date he died. The trust will file a return for after, however:
If the trust has already been in existence and files with a calendar year = a tax year, then yes, the trust needs to file a 2024 return by April 15, 2025. However, if the trust came into existence with your father's death, and does not have an established tax year, you may be able to choose a different tax year, which would change the deadlines. I can't tell you the advantages or disadvantages of doing that, you would need to ask your CPA.
To file his personal 1040, you need to be appointed the executor or representative of his estate, this may require a trip to the probate court in your county. To get any refund deposited in your name (rather than the estate), you file an extra form with his 1040 (form 1310).
https://www.irs.gov/newsroom/filing-a-final-federal-tax-return-for-someone-who-has-died
To file the trust return you need to be authorized (hopefully you are the trustee already). If not, you may need to see an attorney or the probate court again.
