I have a small area of 15 feet by 10 feet and am thinking of doing three 3 foot by 8 foot beds with about 2 feet in between each. But I could also do two large beds instead like two 5 foot by 8 foot beds. Are there any advantages to either? I just want to make sure that the space is usable.
Think about what you're going to plant and how much you want to plant. Come up with a list of everything you want to put in your garden and sketch it out roughly to scale to see if the design will work better for you. It's easier to make changes on paper before you build your beds.
A few years ago I read up on Square Foot Gardening by Mel Bartholomew but never implemented it. The book has consistently been one of the most popular books on gardening since it first came out and it's practiced by many home gardeners around the world. I reread the book and might build my raised beds this fall.
In the SFG System everything based is on a 1 square foot planting area. He recommends planting beds to be 4'x4' because you can easily reach into them and walk around them. 3'x3' is a good size for kids. Some people build longer beds but it's important to have one dimension not be bigger than 4' so that you can comfortably reach any plant. If the bed is only accessible from one side it shouldn't be more than 2' wide.
As for the length, having it too long means you have to walk a long way around to get to the other side. Aisles are a very important part of SFG and the new book emphasizes that. You need comfortable space to not only walk around but to work around. You'll be spending time kneeling and squatting and need space to do so without disturbing the plants. He recommends 3 feet for aisles but for years in the beginning he had his beds spaced 12" apart.
One of the reasons Bartholomew came up with the SFG system decades ago was because conventional row gardening was so wasteful. For example, seed packets have instructions like plant rows 3' apart and thin to 6". The row spacing is an artifact of production farming that uses heavy equipment. When working in raised beds you don't have to think about rows. If plants are thinned to 6" they only need 6" around them which means you can plant 4 in one square foot. Plants that are thinned to 4" can fit 9 in a square foot. Thinned to 3" you can fit 12 in a square foot.
Large plants take up the whole square foot. A few plants need more than 1 square foot.
Vining tomatoes fit in 1 sq ft provided it is trellised and side shoots are pruned weekly so only the main stem is left.
I would recommend the book even if you don't follow the whole square foot gardening system as you can still learn a lot from it for raised bed gardening in general.
Also... Don't limit yourself to having all your beds the same size and shape. Be creative and your vegetable garden can be a very attractive part of your landscape. Here's one example from http://blog.kittykono.com/2013/06/my-little-raised-bed-garden.html
I've always liked vegetable garden designs that become little outdoor rooms. Here's something I quickly threw together in a 15'x10' space. There's more square footage of beds than the 2 or 3 rows you had mentioned.
I think it depends what you're wanting to plant. For instance a package of bush beans I happen to have next to me says "Plant in rows 24 inches apart". So if you have 3 rows, 36 inches wide, I think you could get 3 rows of beans. But if you had 2 rows 60 inches wide, I think you could get 4 rows of beans.
However, if you planted pole beans, the instructions say "Plant in rows 48 inches apart". In that case, I don't see more than one row per bed. So, you'd get more rows if the beds were 3 ft wide.
My tomato pack also says rows 48 inches apart. Cucumbers says rows 60 inches apart.
Basil is 12 inches apart. Corn rows are 24 inches. Peppers rows are 24-36 inches.
Onions are 2-4 inches apart, so then it would be an area calculation. 2x5x8 = 80 compared to 3x3x8 = 72. You have 8 more sq ft to plant onions if you had 5x8 beds.
If you're going to be planting tomatoes and cucumbers mostly, I'd go with the 3 rows 3x8. You'd also have more room to work between them. You'd have room for a wheel barrow.
If you did three 3 ft beds, you could push the 2 outter rows all the way to the sides of the area and increase the path-width to 3 ft. If you did two 5 ft beds, you will need a path on both sides of the bed, which reduces the path size to 5/3 = 1ft 8 in. That's not very wide.
My 20'x20' garden is laid out like the 2nd diagram (domino-grid looking); I will be putting a keyhole garden in the center. I originally had rows, but I get lots more linear feet of growing space with this design. Is enclosed with 6' high trellis fence for vining plants; a gate at the center (as shown here) that opens to my chicken pen. When ready, I just let the chickens in and they keep it weeded for me. I also have a gate at top right corner (seldom use, but gives access w/o going thru chicken pen. Love it!
I just built 3 foot wide beds with a fence around the two beds. Since I can't reach it from both sides, 3 ft is almost too wide. Think about that stuff too!