This is a very loaded question. When I talk about "styling" bonsai I use that term basically synonymously with "developing" bonsai. When I started with bonsai, I came across many videos of folks picking up nursery stock and chopping up/heavily wiring the tree - and voila a "bonsai". As far as I'm concerned, this doesn't work. I for sure cannot create bonsai this way and don't waste my time/material trying anymore. I have done some truly maniacal things to many a spruce attempting this "bonsai in one go" method. Unless one purchases a very expensive "finished" bonsai tree that has been grown for bonsai over a long period of time, we will need to "develop" our material over time. Pruning will play a huge role in this process.
There are essentially three areas of the tree that will require pruning over development and the life of the tree - (i) roots, (ii) trunk; and (iii) branches. Two things to internalize right from the start - (i) bonsai takes a long time to develop (don't be the kid in the car asking "are we there yet"); and (ii) your tree will not look good for many years as you develop it. The second notion was very hard for me to swallow at least.
Pruning and growth are the essential tools of bonsai development of roots, trunk and branches - we will always be trying to strike a certain balance between growth and pruning depending on the specific stage of development. Starting at the roots, we are always trying develop our nebari - the tree's lateral root flare - from day one to the end of the tree's life. At the correct time, the tree is repotted, roots cut back and arranged laterally and the tree replanted. The roots are allowed to grow for a season or 3 and then we do this again. Over time, the roots ramify and form the base of our tree. These aren't just roots - they are providing the initial girth, taper and even movement of the tree.
Trunk and branch pruning will depend on the material we have in hand and the goals in mind. And there is an order of operations of sorts here - trunk development comes before branch development. Of course, we will need branches to grow leaves to perform photosynthesis and feed the tree, but we are not necessarily "styling" these branches if the trunk is still in early development - they may not be part of the final design of the tree.
Remember, we are ultimately trying to create certain proportions of the tree on a smaller scale than what we see in nature. The tree will absolutely not develop these proportions unless we intervene - the trees will naturally try to grow giant. There are many "rules" of proportion that you should read up on, but these are not rules in the sense that something bad happens if you break them, but rather these rules are observations of what the general public finds aesthetically pleasing. For example, a 1:6 ration of thickness at base of trunk to height of tree. So if I want a 15" tree, I will need a 2.5" base. If I have a stick in the pot seedling, I will arrange my nebari and let the tree grow for some years to get the base I need. With deciduous trees at least, when I have my base, it's time to prune the trunk - 1/3 of the size of the final tree is a good rule of thumb. My very tall shoot is now a 5" stump. A new leader is grown, perhaps movement is added here by angling the new leader, and this is allowed to grow again until I have the next section of my trunk with some taper. It will be cut back again. As Brent from Evergreen Garden Works says - we are growing trunks at this point.
Back to notions of balance - trunk thickening is all about free growth. More leaves = more photosynthesis = more food = fatter trunks. It's as simple as that. Pruning takes away the source of food and slows growth. On the other hand, big chops leave big scars that take longer to heal. So you will ultimately need to trade off between faster growth and smaller scars/better taper & movement. This is one of the things you will need to play around with and find your preferred way.
Once the trunk is further along, we can start keeping branches we like. Again, notions of proportion will dictate. Lower branches are generally thicker than higher ones. To maintain proportion, we will prune what we want to slow down and allow what we want to thicken to grow. Pruning will be used to also create taper and movement in the branches, much like we did with the trunk.
Pruning will be used to create ramification in the branches. This not only looks great, but it also causes the leaves to reduce in size. The basic process is keep the branching diving into two. If there is a place where the branch splits into 3 shoots, cut one off. Prune the other 2 back to 1 or buds and let it ramify again. Repeat this season after season until you are satisfied with the branch structure. Wire can also be used to add movement to the branches and sub-branches.
At some point, we have a tree that we are no longer attempting to really develop further - instead, we want to maintain the structure/style that we have already developed. We will try to prevent shoots from extending altogether by pinching and similar pruning processes to limit growth so things don't overgrow lengthwise or widthwise.
There are a number of questions that need to be answered before putting any of this into practice. For example -
(i) Is the tree deciduous, conifer or tropical? Each will have their own general way of proceeding along the bonsai development path.
(ii) What specific type of tree are you working with? Trees will vary in their responses to development work.
(iii) & (iv) What stage is the tree at/where do you want it to ultimately be? If you have a young tree, chances are you need to grow and not prune. Perhaps you found some good nursery stock or splurged on some better pre-bonsai - you could be further along in the process, maybe you have 2/3 of a tree and need to start on branching and re-do the top 1/3 of the tree.
On a final note, pruning in combination with wiring (and sacrifice branches) will be used to create a "style" for the tree. The trunk on a formal upright could be cut at 1/3 the final height with one branch wired straight up for the next section of leader. If more movement is desired for an informal upright, the cut could be made in the same place with the leader wired on an angle to create more of an "S" shape. Allowing multiple leaders to grow from the cut and then pruning those back to ramify will create a broom.