The majority party in the House of Representatives gets to appoint the Chairman of every sub-committee in the House, there are many of these. These chairmen are all Republicans, but will soon all be Democrats, every one of them. The ruling party needs no excuse to replace them, and all chairmen are always appointed from the ruling party.
Each of these Chairmen has the authority to issue subpoenas (these are legal documents that can require the recipient to testify, to turn over records (like emails), and so on, refusing to do so is a crime that may result in being jailed for Contempt of Congress).
It is the job of both the House and Senate, independently if they like, to oversee government operations, conduct inquiries, get testimony under oath (lying under oath is perjury, a felony that can be punished by five years in prison).
YES, the House has the authority to demand from the IRS the Tax Returns of Donald Trump, pursuant to any number of investigations they might undertake to see if he is taking emoluments (payments from foreign countries) or profiting from his office or making decisions to benefit his own business, family, etc. The House does not need ANY excuse or evidence or permission to start such an investigation: They are in charge of oversight.
Further, the House chairmen are free to re-open any investigation their predecessors have put aside. So they can re-open investigations into Russian meddling in the 2016 election, or 2018 election. During this investigation, the Democrats requested something like 64 subpoenas of records and testimony; the Republicans denied every single request. Well, every single one of those subpoenas will now be issued, the witnesses heard under oath before a Democratic Chair, and that includes Trump's children, friends, business records, business employees, tax returns, etc. Unlike the Republicans that excused blatant lying by some of these witnesses and allowed some (like Trump's son) to simply refuse to answer, a Democratic Chair can compel testimony under penalty of Contempt.
Further, for most of this testimony, it can be public if they wish, or be behind closed doors but all or some of the testimony revealed to the public. That is also within the purview of the Chair (with exceptions for some classified material).
Although the House cannot unilaterally pass any law (they can introduce one, but both the House and Senate must approve and the President must sign; or if he vetoes, be overruled by 2/3 of both).
But this power of investigation into corruption, self-dealing, foreign influence, and so on is actually a very big deal, and the results of the investigation can result in criminal charges. The House cannot be restrained or gagged by either the President or Senate, it is subordinate to neither one.
Finally, only the House, after investigation, can bring impeachment charges against the President (actually they can impeach any executive or judicial branch federal government employee; including judges and Supreme Court justices). These charges must then be heard by the Senate, which may vote to remove the offender from office. It is unlikely they will, especially with a Senate friendly to the President, but you never know what they might uncover that would change the minds of at least some Republican Senators. So yes, there is a remote chance that even a "New President" is the result.
P.S. Also, all spending bills raising revenue must originate in the House; including laws raising or lowering taxes (the Supreme Court has held). Neither the Senate or the President have the right to raise or lower taxes.
P.P.S: All laws must be passed by both the House and Senate, by a majority in each. Typically the President must also sign a new law, but if the President Vetoes, then a 2/3 majority of each Chamber (House and Senate) can override a Presidential Veto and pass the law anyway. So Democrats can now block any NEW law. This also means they can prevent the repeal of any law, since repeal is treated the same as new law. So they will likely not allow any changes to the social safety net; Medicare, Medicaid, Obamacare, Social Security, etc. There has been talk of re-allowing health insurance companies to refuse to sell insurance to or to charge more to people with pre-existing conditions (e.g. a person has asthma, so they can be denied insurance even against unrelated diseases like cancer or tuberculosis). The House won't allow any such repeal or new law to pass. Also, since the House can veto a budget or other allocation of funds to build the Wall against immigrants spanning the border with Mexico, they can ensure that doesn't happen.