Here is my 2 cents on the subject.
The 7x6 board is solved mathematically, and proven a first-player win, presented by L.Victor Allis in his master thesis: "A Knowledge-based Approach of Connect-Four (1989)" The 8x8 board however has not to my knowledge been subject of the same research, and amongst players on IYT and GT it has often been discussed if the first or second player has the advantage. McStack has pointed out, that the first player is the first being able to make a weak move and therefore the second player has an advantage with defensive play. My personal feeling based is on this, and the fact that the second player have more possible winning combinations, are that second player in most games, has the advantage. First player can in my opinion, by playing "perfect", only achieve a draw in many openings, with time I´d might even argue that the 8x8 board is a tie!
For tournament strategy I´d prefer to play 2 games one of each color, against each opponent, this opens for a more fair and strategically challenging 4iar tournaments.