One of the things that some new cubers get caught up in early is trying to turn as fast as they can. It's one of the first things that getting a better cube is going to do for you, because the better mechanism will allow for faster turning. While turning faster is important, it's equally important to know when it will be the most effective.
When you're nearly done with a solve, there are fewer remaining positions to consider, and as you get more experienced you're going to know what those positions are probably going to be. However, at the beginning of the solve, there are a lot more possible positions that the pieces you're looking for can be in. That's why looking for pieces while you're doing other moves is important. Sometimes you will spot a piece you need later right as you're about to do a move, but it's something that's going to get moved by what you do. Do you know where that piece will end up at the end of the algorithm that you're doing? Alternately, you may intentionally look at a piece in a location that's not going to be affected by what you do, and think about where that goes and what you're going to need to do next.
If you're using the Beginner's method, look-ahead may not seem important yet because after the first four edges (which hopefully you mostly covered during inspection) you're going to look for the first layer corner pieces one at a time to finish the first layer. However, if you have time to look for the second corner while you're doing the first one, there will be less pausing between moves. The same would go for the middle layer edges. As you put each middle layer edge in, since the algorithm should be muscle memory, you should be looking around the cube to find where the next piece you need is and where it should go.
If you're using F2L as part of the CFOP* method, where the middle layer edges go in at the same time as the first layer corners, look-ahead is the most important. If you just went "What's CFOP?" here's a quick refresher.
*In the CFOP method, we have 4 stages.
We have the first four edges, commonly called the Cross (That's the "C" part.)
have the first layer corners going in with the middle layer edges,
which completes the First 2 Layers (which is the F of CFOP, referred to
we Orient the Last Layer(OLL) which is where we turn all the last layer
pieces the correct way, and then Permute the Last Layer(PLL) which is
where we move around all of the correctly oriented last layer pieces to
their correct locations.
While the OLL and PLL moves primarily require quick recognition and fast execution, the recognition is relatively easy because you're only looking at the last eight pieces, and execution speed will just come from practicing the moves over and over. The speed of the F2L totally depends on your ability to look for two matching pieces at a time without having to stop, and represents the slowest part of the solve because you're having to react to a pair at a time. You can automatically ignore any edge that has the last layer color on it (easier to say than to do), but you still have to mate up a corner piece with its adjacent middle edge and then put those pieces into their slot. The corner can be oriented in three different ways, in eight different places, and the edge can be in one of eight places. So, as you're inserting a corner-edge pair, you need to look for the next pieces to put into place. As you get more advanced with F2L, you may opt to place your first pair in one of the back slots so that you can better survey the open slots in the front and the unplaced pieces in the last layer.
F2L represents a serious improvement over the beginner's method once mastered, because placing a corner usually takes 3-4 moves, but could take up to 6, and placing a middle layer edge afterwards takes 8 moves. Placing the corner and the edge together in the worst of circumstances maybe takes 11 moves, but usually takes 7-9. On average you're going to save around 20 moves by switching to F2L, but that's only going to save you time if you're accomplishing that at speed, and that's only going to happen if...
Anyone? Let's not always see the same hands.
That's only going to happen if you're looking ahead.