I have been folding origami more than 20 years. During that time I've come up with some important strategies for making great origami models with paper. A nice thing about origami is that there are a lot of simple models that a person can do without much experience. However, as you get into more complicated models, there are some tricks. This is a short list of tips that I hope you will find useful.

Always Review the Diagrams and Familiarize Yourself with the Symbols Every origami book devotes a few pages at the beginning to the symbols in the instructions. When you start out for the day or get a new book, take a moment to refresh your memory. I find that taking the time to go over the basics puts me in a more attentive state of mind.

The First Paper Folds are the Most Important When you first begin an origami model, the first five or ten folds really make the foundation of the model. As the number of folds increases, models tend to get more out of control. It seems like this isn't linear. The more folds pile up, the harder it gets if something you did earlier wasn't quite right. When you first start, pay special attention to making edges line up, and also to make sure that your creases meet the corners precisely.

Choose the Right Kind of Origami Paper This may be one of the most important choices you will make when you start an origami model. Here are the most basic types and what they're good for. Here are some basic types.
Foil-backed Paper A thin sheet of paper glued to a sheet of metallic foil. This paper will hold a fold well and is good for making sharp edges. Also, it is good for more complex models because it compresses well and stays put. One problem with foil-backed paper is that you can't smooth out mistakes as easily. Also, if it gets dented or dinged, it's more permanent. You can buy this sort of paper at stores that sell origami paper.
Solid-Colored, Thin Paper If you are doing models that aren't super-complex, but aren't exactly simple either, this is a good kind of paper to choose. The thinness is good because if you're making something where there are lots of fold (and thus lots of layers) thin paper is better. Also a lot of more complicated models have features or details which are showed off better with plain papers.
Paper with Texture I love using this sort of paper for more simple models. Textured paper has more volume, and also adds a lot of character. Designs in the paper add interest.
Scrapbooking paper Since scrapbooking has become so popular, there has been an explosion of kinds of paper. Textures, prints, designs, and thicknesses that an origami enthusiast would only have dreamed of years ago are available now.

Folding in the Air Experts caution against folding models against a surface like a desk. They recommend using your hands as much as possible.

Take Breaks While Folding Origami A lot of times when you're doing something, especially if it's fun or needs a lot of concentration, you might forget to take a break. Taking a break here and there can help you clear your head. Also it's a good idea to pause from time to time and check your work. Sometimes you might make a mistake, like folding something up instead of over. If something doesn't look quite right you may find that you missed a step. Some people also take time here and there to check out the next few steps. They can see what changes are coming and can then plan their next moves. More experienced origami model makers will also see where creases should be sharpened.

Pay Attention and Enjoy the Experience I love to observe the process of an origami model taking shape. Basic shapes become more complex. Pieces of the model are folded to become limbs, body parts, or wings. Seeing the overall pattern emerge as you work on origami models and as you fold the paper can be very rewarding.

I hope you have as much fun folding origami out of paper as I do.