Apparently, this is known since 1988, when Victor Allis, from the Department of Mathematics and Computer Science of the Vrije Universiteit in Amsterdam has published his Master Thesis: A Knowledge-based Approach of
Connect-Four — The Game is Solved: White Wins.
Apparently, despite the huge number of possible positions, the game can be fully analysed, and the first player can always win — with an ideal play, obviously.
From the long series of analyses, I’ve selected a short one from the chapter § 3.1. Useless Threats. Diagram 3.3, black to move (the paper mentions the colors as white and black, not yellow and red):
White has just played his 6th move and it is Blacks turn. Black already has his threats at b2 and f2, and, having read the former sections, Black knows that eventually White has to play b1 or f1 (if Black can prevent White from winning before that time). So Black tries to find possible other threats White can compose. There are of course the vertical possibilities, but these are always easy to refute.
Diagonally White can achieve nothing, without a square in the range b2-b6 or f2-f6. So Black does not have to worry about that either. Horizontally the same is true, except for the group a1-d1. This group can be completed without using a square in b2-b6 or f2-f6. If White takes a1, Black has to play b1 and then White can get b2. So to destroy White’s last hope, Black plays a1. We now know that White cannot complete groups either horizontally, vertically or diagonally. Moreover we know that White will be forced to play b1 or f1 eventually. Therefore we know that Black will win.
From the too many Connect Four apps for Android, I’d recommend the following: 4 in a Row, by Ultima Architect Inc.; Four In A Line Free, by AI Factory Limited; Four In A Row, by OutOfTheBit ltd; and for added puzzles, Four in a Row HD, by Alberto Hernandez.