Saying that they can investigate things is a little bit vague. Specifically, they can do things like subpoena people, have them be held in contempt, etc. Though, since they’re not actually a court, the exact rules around all of this are a bit squishy and could prove to be less than actual if someone ever fought it to the Supreme Court. They like for people to do things voluntarily, because that saves them from having to invoke their uncertain powers in this realm.
When they bring someone in to testify, they can choose whether to do that publicly or privately.
The different oversight committees have specific powers that allow them to do their job of oversight. This can mean anything from seeing banking records, to tax returns, to FBI reports, etc. Traditionally, the chairman (majority party) and the ranking member (minority party), could issue a variety of requests for information from the Executive branch and expect to get a response. Under Trump, last I heard, they were only processing requests that had come from the chairman.
Each Committee can publish a report on their findings. They can release information to the general public - testimony that was taken behind closed door, etc.
One assumes that they can throw a wrench into the gears of any particular branch of the Executive branch through their Oversight powers. Potentially in direct ways (?), but certainly at least by harassing them with frequent, large requests for data and calling people in to be harangued on a regular basis. In general, they don’t have any powers except to see stuff, but I wouldn’t be surprised if they’re part of the system in at least some way - they have to stamp something every once in a while or whatever.
The House would have standing to prosecute the President, a State, or really anyone for anything under which a court decides that the House has standing.
From Wikipedia: “Under the Twelfth Amendment, the House has the power to elect the President if no presidential candidate receives a majority of votes in the Electoral College. The Twelfth Amendment requires the House to choose from the three candidates with the highest numbers of electoral votes.”
The Speaker of the House can become the President, via the line of succession. That’s not something they can force to have happen, but it’s technically part of the rules and doesn’t involve the President.
From Wikipedia: “The Speaker is also responsible for overseeing the officers of the House: the Clerk, the Sergeant-at-Arms, the Chief Administrative Officer, and the Chaplain. The Speaker can dismiss any of these officers. The Speaker appoints the House Historian and the General Counsel and, jointly with the Majority and Minority Leaders, appoints the House Inspector General.”