The White Cross
Similar to a lot of things in technology, solving the Rubik's cube is best done by breaking it down into a series of smaller problems and progressively working towards completion. In solving the cube we will start off with the white face and progressively work towards the yellow face layer by layer. The white face will be solved in two steps, create a white cross, then in step two we will insert the white corners.
In creating the white cross, it is essential that the white edge pieces are aligned so that their coloured side matches up with the corresponding coloured center pieces as in the diagram below. At this stage we don't need to worry at all about any of the other pieces so I've greyed them out.
See if you can do this step intuitively (ie without using any algorithms). This is a good way to help build up your skill and familiarity with the cube. If you get stuck however read the advice below.
The general strategy which is used is that once pieces have been placed we want them to stay in place. It's ok to move them out of place in order to place another piece as long as it is temporary and that piece is put back after.
Whilst creating the white cross we have a lot of flexibility as we don't need to worry about disrupting any of the other pieces. A good approach is to pick a particular white edge piece and move it into place, then the next, then the next then the final one. Don't try to place all of them simultaneously (though as you build your skills you may get to the point where you do line several up to be placed simultaneously).
Here is an approach that works well:
Depending on whether your white edge piece ends up with the white panel on the yellow face or not there are two ways to slot it into place.
This one is the easiest. Simply line it up directly above where you want to slot it in then rotate the corresponding side face 180 degrees and you're done.
This one is a little more difficult, but barely. Instead of one 180 degree rotation we instead do three 90 degree rotations. We are going to slot the piece in rotating it through two of the side faces in order to twist it around so that the white panel is oriented the right way.
Line up your piece directly above where it should go then rotate it 90 degrees to the left.
This move works just the same if you rotate 90 degrees to the right as well.
Although we are starting with the white face there is nothing stopping you from starting with another face. The cube is symmetrical on all three axis so it really doesn't matter which face you start from. Speed cubists often look at the whole cube and start with the side which is quickest to solve which can save you a good few seconds with a bit of experience. Practice creating the cross (and the other steps as you learn them) starting from faces other than the white face so you can easily adapt.
Another good thing to try is to see if you can place the four edges without reorienting the cube. That is, if you start off with the red face pointing at you (for instance), keep the red face pointing at you at all times. This will help to build your mental image of the cube (making it easier to visualise how the later, more complex stages work) and also help shave off precious seconds that you don't need to waste rotating the whole cube.