Does a mortgaged home purchase impact your federal U.S. tax return in any way?
In the US you can take a standard deduction or itemize your deductions. Mortgage interest and property tax tips the scales in favor of itemizing deductions for many people. So the answer to your question is, it depends. If the sum of your itemized deductions exceeds the standard deduction then you pay less federal income tax by itemizing your deductions. If you wouldn't itemize deductions if not for mortgage interest and property tax then buying a house would impact your federal tax return favorably.
Since the standard deduction was recently increased and the state and local tax deduction was capped in 2018, fewer people benefit from itemizing and the benefit is reduced among those that still do. The impact of the tax law varies by state, since some have much higher property tax than others.
There have been federal first-time home buyer credits in the past, but not presently.
Possibly - you can deduct mortgage interest from your federal taxes if you itemize. In order to benefit from itemizing, though, you need have (for 2020) more than $12,400 in deductible expenses, or $24,800 if you are married and file jointly. This is not as common as it was before the recent tax changes, and typically requires either a large mortgage and/or other deductible expenses like charitable contributions, state and local taxes, or large medical expenses.
That said, don't buy in to the hype that getting a mortgage is "wise" because of the tax benefit. If you can itemize, you effectively lower your interest cost by you marginal tax rate, so a 4% mortgage for someone in a 25% bracket effectively becomes a 3% mortgage. Or as my father always said - "Don't send the bank a dollar just to keep from sending the government 25 cents". Buying a home can be a wise investment, but overspending on a house just for the tax break is not wise.
In a comment, you clarified that you were specifically asking about a first house:
I was hoping for a more cohesive overview of what tax elements come into play for a new homeowner
At the federal level, there are no tax benefits to buying your first house. You get the same benefits as anyone else who owns a house with a mortgage. This has been a change from the days when the Housing and Economic Recovery Act offered a tax credit to first time buyers from 2008 through 2010. Some people who may wonder about a first time home buyers tax credit may be remembering that program.
However, some states have specific programs or tax benefits for first time buyers. Some of these programs can essentially trickle up to your federal return. It's hard to describe if this applies for you or not, because we don't know your state.
Some of those state programs operate as tax credits, but in other cases the link to your taxes is indirect (i.e. some programs will subsidize your down payment with additional cash if you meet certain criteria, which is effectively tax-free money in the year in which you buy your home).