The only way to remove a sitting president through legal action is impeachment.
This requires the House to pass the charges against the president, i.e. articles of impeachment, and the Senate to hold the trial based on the articles and decide the outcome. Impeachment is a political process, not a legal one. Congress, if it has enough support, can toss a president out for pretty much any reason, as long as they can justify it to their electorate.
To gauge how such an effort might go, let's consider two recent events regarding this: the resignation of Richard Nixon, and the impeachment of Bill Clinton, and the fallout from both actions.
Nixon resigned rather than face the even greater disgrace of certain impeachment, essentially for lying about covering up the Watergate breakin. In truth, breaking into the Dem's campaign headquarters was a silly stunt that yielded nothing of real value, and did not affect the outcome of the election. McGovern's campaign was such a mess that he was going to lose no matter what. Relevant factors were Nixon's repeated public denials of having any knowledge of the breakin or the aftermath, shown to be false when the recordings of his conversations showed that he did authorize a coverup. Aiding the impeachment push was the disposal of the even less likeable Spiro Agnew... no one wanted him as president.
The Nixon resignation had a far reaching impact on the nation. The US went through a period of political turmoil in the mid to late 1970's, and looked very weak internationally. The fallout from that weak stature includes the Soviet invasion of Afghanistan and Iran taking the US embassy staff hostage. Not the only factor, but a very relevant one. Note that when a strong president was elected (Reagan), the Iranians coughed up the hostages very quickly, one minute after the inauguration.
And then there is Bill Clinton. In his second term, the House (under republican control) pass an article of impeachment based on Clinton lying under oath. Despite a crime having been committed (Clinton was disbarred from legal practice as a result), the Senate decided not to impeach.
Relevant factor in that was the perjury under question was more an embarrassing personal matter, rather than something that could have seriously affected the nation. The Clinton perjury was a fairly low key affair, not highly publicized, and certainly not repeated by him in numerous public statements. Also, most legislators remembered the fallout of the Nixon resignation, only 15 years in the past, and really didn't want to subject the country to that ordeal again. That would come back to haunt Newt Gingrich, speaker of the House who pushed the articles of impeachment through... he became politically insignificant after that.
It is possible that the new House could pass articles of impeachment against Trump. However, unless the articles outlined credible charges of serious malfeasance, the current Senate is unlikely to subject the nation to the chaos and weak international posture that followed the Nixon resignation. There just isn't sufficient reason, and the economy and employment situations are better than they've been for quite some time.
Even if the Dems recapture the Senate and have the capability to see an impeachment through, they would be well advised to proceed carefully. When the nation finally emerged from the shadow of the Nixon resignation with the election of Reagan in 1980, the Dems found themselves marginalized for quite some time after that. Economically, the late 1970's were not a good time for US citizens.
The electorate will hold the other party responsible for the outcome of an impeachment effort. And the outcome of a Trump impeachment will likely be declining economic conditions and problems internationally for quite some time... a Pyrrhic victory that could turn a lot of voters against the Democrats for at least a decade.
Yes, this is a very slippery slope to follow... if you consider what happens next.