Do you mean carpet, or an area rug? The latter is a useful way to pull together a functional grouping of furniture -- I'd definitely use one rug in the living room part of the space, so that sectional, additional chairs, and so forth all fit onto it (at least front legs). Whether you want a rug in the dining area is more a matter of personal taste -- the table and chairs already define a grouping/zone pretty clearly, so it's not that important in that sense. The main time people suggest them is when your table is similar in tone to your flooring, in which case a rug can help in creating some contrast/separation.

I'd put an area rug in the livingroom - 9 by 12 is the largest, but maybe a smaller one would do. I don't think you need one in the dining room and it might be problematic with the long table and the counter stools. Maybe you could get the livingroom one first.

I would go as large as possible in both spaces without it looking strange. A large rug makes a room or area look larger, whereas a small rug does the opposite. The style and tonal values should be similar, and I would be sure to repeat some of the colors from one rug in the other.

Two rugs that can correlate but are slightly different to define the two different spaces. This helps to divide up open concept areas and effectively keeps things from looking cluttered.

First, do some measuring. Table top + 4 feet all around. Sofa width + 2-4 feet at each end. This will tell you what you're looking for in rug sizes. If those measurements don't fall within standard rug sizes, go up to the next size. I like finding two rugs in a similar style (oriental, IKAT, geometric, floral) with a uniting color theme. Here are two examples from the same company with similar shapes and colors, yet different. Companies often tend to use the same yarns in different rug styles, so that is often a easy way to match the colors in two rugs.

Oriental Weavers Brentwood 001H9 Ivory RUG · More Info

Oriental Weavers Brentwood Ikat Ivory/Blue Rug, 7'10"x10' · More Info