I should start by confessing that I'm not a perfectionist. I'm a "good-enoughist," which means that I rarely concern myself with such issues as things being perfectly level. If things like that matter to you, then you'll need to factor that in when or if you try my bed-building method.
As with anything in gardening, location is critical. Does the proposed bed location get sufficient sun? Is there a water source nearby, or will your hose reach that far? How high do you need the bed to be? How large? Once you know the answers to those questions, you'll know what materials to buy, and how much.
I recommend ALWAYS checking to be sure there are no buried utility lines where you plan to place your bed. In the United States, most states have a special phone number to call before you dig. The utility companies will come out and mark the gas/water lines, etc., and you can avoid future headaches by not placing your beds over those lines.
I use what I can afford, which for me means buying landscape timbers when the big box stores have them on sale, and bringing home as many as I can fit in my car at one time. The big box stores will cut them into whatever lengths I need. I use 4inch exterior screws to connect them, and short lengths of rebar to stablize the walls.
I also watch the sales for bags of inexpensive store-brand soil. And you'll need cardboard/newspapers, as many as you can stockpile. Yard debris, tree branches, and other brush can come in handy as well (more on that later). I add soaker hoses to each of my raised beds, to make watering easier.
Tools: The only tools I've ever needed to use are a cordless drill and sometimes a hammer.
I drill pilot holes in the landscape timbers for the deck screws. One time I had to chop some tree roots that were in the way, and dig a little trench into the ground for the bottom layer of timbers. That was on a very irregular piece of ground, and it was the only way to have any semblance of a level wall.
If the site is overgrown with brush or tall grass, I might weed-whack it or spray a chemical herbicide on it first, but usually I just pick a spot and start working.
Framing the bed:
I start by placing the bottom layer of landscape timbers, just to verify for myself that this really is the location I want. I use landscape timbers, so my beds are angular rather than curved, but within that limitation I can make whatever shape I want. I usually stick with squares or rectangles.
Once I've confirmed that this is the place, I place several layers of cardboard or newspapers under the landscape timbers. This serves two purposes - it kills the underlying grass and it provides a weed-block to help prevent weeds from coming up through the edges of the bed. Bermuda grass is the weed I most often have to guard against, and it finds any little open crevice and follows it to the sunlight.
With the bottom layer in place, I start adding timbers. There are 2 ways to do this, and I still haven't decided which way I prefer.
One method is what I call "even edges," where I place the timbers directly on top of each other. This gives me uniformity, but it also keeps the walls of the beds separate from each other, as you can see in the photo below.
The other method is what I call "interlocking edges," and while it connects the walls for greater stability, I always wind up with uneven corners and occasional gaps.
I use at least 3 deck screws for every timber, sometimes more on the bottom layers, for stability. I try to space them evenly across the length of the timber, and I try to stagger the location from one timber to the next, because if I used the exact same location on every timber, the new screw would be hitting the head of the screw in the timber below it.
I make my beds at least 3 timbers high, and usually 6 timbers high, although I do have a couple that are shorter.
Once the walls are in place, I use short lengths of rebar to stabilize them, placing 2 or 3 of them on both the interior and exterior of each wall, and using a small sledgehammer to pound them into the ground until they are level with the top row of the bed. It's best to do this as soon as you have the walls up, even if you don't plan to finish the bed at that time.
Filling the bed:
Before joining All Things Plants, at this stage of my bed-building, I would take the rest of my cardboard and newspaper, and layer it on the ground inside the bed, hosing it down once it was in place. Again, this serves to kill any underlying vegetation, and block weeds. Then I would start adding the bagged soil and bagged compost.
But since reading Dave's article on his Hugelkultur beds, I prefer to use a modified Hugelkultur concept. Remember how I said in the "materials" section that yard debris could come in handy? I take grass clippings, old firewood logs that are starting to rot, tree branches and other yard debris and toss it in the bed. This helps me make dirt the way nature does - by decomposition, and it gives me a place to put yard debris that would otherwise go to a landfill. It also lowers the cost of a new raised bed, because I don't need nearly as much bagged soil/compost to fill it.
These two pictures show you the before/after of a bed I built recently. For this one, I framed it out last spring, and spent all summer/fall/winter filling it with yard debris. Then in February or March, I topped it all with newspaper and soil, and planted my daylilies. They LOVE it.
As you get more comfortable building raised beds this way, your creative side can come out and play. Rectangles get boring after a while, so last year I made a two-tier bed.
I liked it so much that I made two more, larger ones, this year. But that's a topic for another day.