The first moves
The best starting move for you is to place a piece right at the center. To counter this move, place your piece on top of it.
Watch out for traps
A common trap is for your opponent to have two adjacent discs on the bottom row, with empty space on both sides. You must move on the bottom row directly next to your opponent's discs, or else your opponent will get 3 in a row with spaces on both sides, which you can no longer block.
Another common trap on the bottom row is when the opponent has two discs with one empty space between them, and empty spaces on both sides. You must move in the middle between the two discs, or directly next to one of them, or else your opponent will get 3 in a row that you can no longer block.
Another common trap is the shape that looks like a sideways "V", with one horizontal and one diagonal leg. Your opponent can make 4 in a row in two different ways, which you cannot block. You will lose unless you manage to connect four elsewhere first.
Keep an eye on diagonals
One of the best strategies in this game is to look at all the pieces and evaluate multiple ways of connecting four pieces in a straight line. Remember that there are only three ways to do so: vertically (which is easy to see and block), horizontally, and diagonally. On the 6x7 board, all horizontal and diagonal lines require the center column, which is why it is so important to occupy the center.
Watch out for diagonals. Four pieces placed diagonally are the most common way in which people often win.
Counting
Often the game comes down to counting: when there are a number of columns that you cannot move in without losing, then you are forced to fill up the remaining columns. If there are an even number of spaces in the remaining columns, then it will be your turn again when they are filled up, and you will lose (unless you manage to connect four while filling up those columns!).
When it comes to counting, spaces on the 1st, 3rd, and 5th row (from the bottom) generally favor Red, and spaces on the 2nd, 4th, and 6th row generally favor Yellow. Therefore, if you are Yellow, try to arrange your traps so that the last remaining spot is on an even row; if you are Red, try to do so with an odd row. This way, if the column in question is the last one to fill up, you will win.