I wouldn't start by trying to tame all 10 of them.
Start with the babies since they will be easier. I would separate them from the parents - you can still keep the 4 babies together. Babies are usually naturally curious and less afraid - without their untame parents telling them people are scary, they should be much easier to work with. Besides, they really should have their own cage after weaning anyway.
You can work with them together, or try to identify the tamest one and start with that one.
Wing clipping is controversial (tails should never be clipped). I remember reading in a bird magazine a few years back that it was "impossible" to tame a flighted budgie, but many people do tame them flighted. Clipping wings also does not make a bird tamer - it just removes its ability to escape. A clipped bird may feel more scared by a person picking it up, because it can't get away. On the other hand, clipping may allow you to have them out more with less risk of them panicking and flying into something and hurting themselves while taming. If you do clip - do not clip too much - many wing clipping diagrams show someone trimming 7, 8 or even more flight feathers, you don't want to clip more than 4 or 5. Clip more and you run a risk of a bird injuring itself falling.
Also, clipping is not something that should ever be done on a permanent basis with budgies. Think of it as a one-time thing that may help keep a bird safer while being tamed. Budgies are strong fliers, and prone to obesity and fatty tumors if they can't get enough exercise. Also, they are very agile fliers - once tame and in familiar surroundings, they are unlikely to fly into windows or otherwise hurt themselves flying.