What happens if it is proven that President Trump has been colluding with Russia during his campaign? Does he go to trial or get impeached? Or does nothing happen?
Only the House of Representatives (sometimes called the Lower House) can impeach a President. The House Judiciary Committee holds hearings and, if necessary, prepares articles of impeachment. If a majority of the committee votes to approve the articles, the whole House debates and votes on them.
The House is pretty much free to decide what constitutes an impeachable offense ("treason, bribery, or other high crimes and misdemeanors" is the formal list in our Constitution). They may choose to or not. A majority ( > 50%) vote of the House is required to Impeach.
Impeachment is equivalent to bringing charges. Removal from office requires a trial in the Senate (sometimes called the Upper House), presided over by the Chief Justice. If 2/3 of the Senators vote for Impeachment, the President is removed from office and the Vice President takes his place.
If the House declined to impeach Trump, nothing would happen...
There's nothing illegal by itself about coordinating with a foreign entity. You can coordinate with anyone you want. Now, if you can prove Trump knew/directed that
- The Russians had hacked the DNC and John Podesta's email
- Were timing the release of some of the more salacious details to cause maximum political damage
- The Trump Administration is somehow proving benefits to Russia in a quid-pro-quo relationship
That would be enough to topple a President. General sentiment would turn easily, and you could Impeach him readily (remember, Impeachment is largely a political process, not so much a legal one).
The problem is you must prove Trump knew all of this and was involved in some way. What sank Nixon in similar circumstances were the Watergate tapes proving he had direct knowledge of illegal activity. Barring something similar to connect the dots, it seems more likely that you could find someone in the Trump campaign who was involved. In that case, the person who was caught would likely face some sort of charges, while the White House would simply deny knowledge and assert that person had acted on their own.