Official investigations into Trump/Russia can happen a number of ways. First, there's Standing Committee investigations, which you see in play now with the Intelligence Committee's investigation. The GOP controls this committee, but there are Dem members, most notably Adam Schiff who delivered preliminary findings Monday. Dems are pushing for a Bi-partisan Select Committee, which would be evenly staffed by GOP and DFL. There are also calls for an Independent Commission, however this commission would also be responsible for coming up with a way to prevent future collusion, if collusion is found, so their work would take longer than the select committee. The last investigation route is through a Special Prosecutor. Thanks Vox for this neat summary.
Here's how a Special Prosecutor (also called Special Counsel) gets put into play: He or she can either be assigned by Congress (and with an over-ride of an inevitable Trump veto with a 2/3 majority), or be assigned by the Attorney General.