Read this tutorial for tips on technique from a beginner level, no-nonsense viewpoint and perhaps next time there will be less frustration and a nice sense of accomplishment when you are done.
While it takes some artistry to make truly unique designs, putting flowers in a vase at the most basic level is something anyone can do with a few tips and specialized knowledge. I'll cover both things in a [hopefully] clear manner.
Step 1: Supplies
You need flowers. The bucket of them pictured was something I put together myself, but most places that sell them have pre-done bunches to be purchased for your vase/container.
And a surface to work on that is a comfortable height. Kitchen tables are usually a nice height, or counters. Be sure it is not a carpeted area, floral work is messy and there are always possible water hazards!
Now go ahead and fill your vase/container with water. Leave a bit of a gap between the top of the vase and your water level to allow for displacement when putting stems in. We'll finish filling it up when it's done.
Step 2: Background Knowledge
First, all tools for cutting need to be sharp.
The reason for this is twofold- you are less likely to injure yourself with sharp tools (less force is needed to cut stems) and you are less likely to injure the stems of the flowers. Flowers are still, technically, alive while in arrangements (until they are obviously dead and brown). They need their vascular structure to be as undamaged as possible. Sharp tools cause less trauma to the stems.
Stems need to be cut before they go into water and should spend as little time out of water as possible.
While stems are out of water, they start to "close up" aka. they start to die at the cellular level. Those cells will no longer be able to function properly and provide water/nutrients to the flower. This will block the rest of the flower from doing well and if they are out of water long enough, significantly decrease the life of your floral arrangement. You have 30-60 seconds of a flower being out of water before you should re-cut the stem.
You should cut stems at an angle.
Stems cut flat can get wedged against the walls of the vase/container and block the flow of water and nutrients just like the stem being uncut in the first place. Cutting stems underwater seems to make no significant difference, so don't worry about that "floral myth."
Clean foliage off that will be below the water level.
I show several techniques for this in pictures. You can also use tools to snip sections off. Foliage should be removed because it promotes bacterial growth in the water. Bacterial growth not only looks bad, and smells bad, but will shorten the lifespan of your flowers. Packets of floral food contain small amounts of bleach to impede bacterial growth, you can add a couple drops yourself if you don't have any floral food packets... or you can do the more environmentally friendly thing and change the water in your vase frequently.
You don't have to use the whole stem of flowers [in one spot, or even in the same arrangement].
Many flowers and greens have multiple side stems growing from the main stalk. You can conserve money and make a more balanced arrangement by separating these and placing them elsewhere. In the last picture, I show a piece of Berzelia which had four stems growing from the main stalk. I have used two in the large arrangement and then took these two to make part of another, smaller arrangement.
With those things in mind, let's start the actual arranging!
Step 3: Greens First, Flowers Later
The way that tends to create consistently better results (for beginners) is the "greens first" method. Flowers first tends to be much more difficult and often results in lopsided arrangements. And when going with whatever, there is a serious danger of arrangements that look thrown in a vase, not arranged (which, maybe is what you're going for, but I'm not covering that here).
Traditionally styled floral arrangements are creations meant to be viewed from all-around. 3-sided and 1-sided arrangements also exist and can be created with this method. I'll touch on that briefly later. For now, we are making a traditional, all-around arrangement.
I have eucalyptus of willow and parvifolia varieties and leucadendron 'discolour'. My flowers and greens are located in the bucket by my feet, but you may also choose to place them flat next to your vase and work that way. It involves less bending and as long as it takes you less than an hour to create your arrangement, no flowers should start to wilt (...unless it is over 80 degrees where you are working, exceptions to every rule). You could also put the bucket on the table next to you, but it might impede your view of the arrangement.
Clean foliage off and add stems to create a pleasing silhouette. You want something nicely rounded and be sure to twist your vase to check all angles. If you run out of greens before you have filled things in as full as you would like, go back and change where things have been placed (or go buy more greens).
You don't want things too tall or too short. A good rule of thumb is no higher than twice the height of the vase and I find 1.5 to be generally pleasing. if the greens are too short, they won't support your flowers and you will have to "fight" with things to stay where you would like them.
This is supposed to be an enjoyable activity, not a battle. Take the extra time to make sure your greenery is well-placed. Sure, no one "cares" about looking at the greens, but it is the structure for your arrangement and if the structure sucks.... well, I don't have to finish that!
Step 4: Start Adding Flowers
In floral work there are three basic groups of flowers: fill flowers, form flowers, and line flowers.
Fill flowers are flowers that don't tend to have much of a visual impact alone. There are lots of tiny flowers ("florets") on a single stem and usually lots of stems come off one stalk. Babies' breath, solidago, wax flower... they add more volume and "fill" empty space.
Form flowers are the flowers with the visual impact, the punch, that tend to draw your eye the most. Arrangements can be made of just form flowers. Roses, lilies, daisies... they engage our eyes and lead them around the arrangement.
Line flowers are flowers that add height to arrangements. Snapdragons, iris, birds of paradise. Without line flowers arrangements tend to be very, very rounded and sometimes lose impact.
Think before you add flowers.
Not only in the general "what does this flower add to my arrangement if I place it X vs. Y vs. leave it out" but also keep in mind the properties of the flower itself. Leave fragile flowers for last. Lilies are prone to losing petals when bumped, roses can bruise, other types of flowers have easily torn petals.
I always start with my filler flower, if I have one, but this may also apply to them and sometimes they get added last.
No, really. Floral work is rewarding in itself and the presence of flowers in a room has been shown to increase mood and relieve stress. If it's starting to bother you, step away for awhile. It's a process and the flowers don't care what water they're in, if they're in an ugly bucket or a beautiful vase.
Step 5: Other Styles
With greens and two iris, I created what is know as a "high style" arrangement. This type of arrangement includes carefully selected blooms for shape and impact and often little or no greenery in unusual containers. As opposed to traditional arrangements it is often purposefully off-center and/or one-sided. Greens in mine have also been put askew and carefully chosen for their impact.
The last is a more "compact" style of arrangement. Often used as a centerpiece, this style chooses to cluster things tightly to the container, though I have broken that up with greens.
There are many, many other styles and types of arrangements available. Go forth and create!