I am trying to model clothes in Blender. I've looked at several tutorials, but the sewing option in Blender doesn't give the desired result. I've been trying it over and over again, but it doesn't seem to work. If anybody could be so kind to check what I am doing wrong, that would be greatly appreciated!
There seem to be a number of issues causing you problems here.
The most important one seems to be simply that the garment is too small for the model - it is quite literally coming apart at the seams :
To resolve this you need to scale the dress so as to give it more fabric to fit around the body.
The above issue is made worse as the collision bounds are such that the cloth isn't actually getting close enough to the surface of the body - the Outer collision setting on the model is currently set to 0.1 meaning that there is a significant gap between the skin and the dress - which further compounds the issue of insufficient cloth to fit! Try changing that to, say, 0.02 for a better fit.
Another issue is the strength of the sewing - currently you have the sewing springs set to 10 right from the start of the animation. This means that the full force is being applied from the first frame, pulling the cloth violently together. Once this overcomes the collision bounds the cloth is below the surface of the body and cannot escape, resulting in fighting between the vertices until the cloth is eventually torn through the model - it's better to apply the forces gradually to gently bring the edges together. This allows the cloth to settle into place rather than being forced together quickly.
The trick is to set keyframes to control the forces more carefully, starting at zero and increasing over a number of frames. One problem with this is that the cloth will naturally fall under the force of gravity before it is pulled together but, again, keyframing gravity to gradually increase can avoid this. In fact, it's useful to start with a weak reverse gravity (eg, weight of 0.5) to that the cloth initially floats gently up, before increasing back to full gravity after 10 or so frames after it has settled.
After this, it should just be a matter of a little bit of fine tuning - such as giving slightly more space around the arm holes and moving it up slightly so that the arm holes can close up around the arms without colliding, changing the collision bounds slightly (reducing Cloth collision settings to 0.5 seems to give be. You could also increase the cloth Quality steps and also collision Quality.
Here's the result :
There still seems to be a bit of overlapping vertices on the shoulders - I think this is due to the arm fabric intersecting with the shoulder. Adding slightly more cloth should do the trick. You could also subdivide the cloth to add more detail for a more realistic fit - although this will slow down the simulation.
Here's the amended Blend 
