Special counsel Robert Mueller’s team will take President Trump’s former campaign chairman Paul Manafort to court in Alexandria, Va., on Tuesday, in a trial with implications reaching far beyond the immediate charges ranging from tax evasion to bank fraud.

The outcome of the trial, which is expected to last three weeks, will help determine how the public perceives Mr. Mueller’s 14-month-long investigation into interference in the 2016 election. A conviction would provide the special prosecutor with momentum as he pushes to...