Sorry to be more specific, we are moving into a new home with hardwoods, so there will be no carpet. Below is the room we will be putting the above couch.

I don't think 8x10 is big enough. What are the dimensions of the sofa? The rug should go under the front legs of the sofa on the right and extend in front of the section on the left to the back of the sofa.

I think the rug needs to go under the fronts of all the sections. When measuring many standard rugs come in 8 x 10 and 9 x 12.

Problem with shag rug is that I have young kids, and I am worried that they will get really dirty

These pictures are from our design portfolio and show some big sectionals and furniture sections with rugs.

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Your instinct is right. You cannot vacuum or clean shag rugs and even without kids, it is pretty gross. You need at least an 8x10 possibly larger.

I really like colorful rugs (Persian and Kilim) any ideas of what you think would look good?

If you just like the style, you can get the cheap oriental/Persian style in polypropylene from pottery barn, overstock, anywhere really. If you're looking for hand knotted, and over-dyed rug would look great.