When I tried to solve it for the second time, I noticed that when the blue face is completely solved, the first layer is still scrambled! What can I do about it?
The answers given so far are correct, but they might be using terms you're not familiar with especially if you've only solved the cube *once*.
The idea is to look at each piece as you put it in place and make sure that it matches the face you are putting it on *and* the neighboring face, or faces.
For example, let's say you are working on the blue face (as you said) and you are going to put in one of the edge pieces. You look at the piece you are going to put in, and see that it's *other* sticker is yellow. So you have the blue/yellow edge. When you put it into the blue face, make sure you put it into the edge spot that is above the yellow center. So the blue sticker matches the blue center, on the blue face, and the yellow sticker matches the yellow center on the yellow face.
Do that for all the edges making sure the center caps match the side color of each of the blue edges you put in.
You will do the corners in a similar way. When you find a blue corner, look at the other two stickers. Let's say you found the blue/red/yellow corner. It needs to be placed at the blue, red, yellow "intersection. That is, the corner where the blue, red, and yellow faces come together.
So pay attention to the entire piece each time, put it in the correct spot, and you'll have a fully solved first layer.
I hope that made sense. If not, feel free to ask questions.
Brian.