I wanted to provide a description of how the right brain resides in the world, but I failed. To put it in a post, I have to use words, and words immediately make it all sound like the left brain.
If you want to get an idea of what your right brain is like, read My Stroke of Insight, especially the chapter "Morning of the Stroke". At that point she was living entirely in her right brain, and she describes what she remembers it feeling like. Of course, anything she writes is suspect because she has the same problem I do, her thoughts must be translated by her (largely recovered) left brain. The description is also suspect because it is a description of a right brain that has just lost its lifelong partner, essentially a fraternal twin, and is both bereft and fighting for survival.
When you open your eyes, the right brain soaks in everything in the visual field. It connects to the world it sees, and knows where it is in the 3D world. It listens to the world. It knows where the sounds come from, and connects the sounds to the 3D world it sees. It feels every part of the world you are touching and adds that to its integrated understanding of the world. The same with smells. All this happens without any effort on your part, and it continues as long as you are awake.
The right brain craves sensory stimulation. The head turns. The eyes scan. The fingers explore. The body locomotes. The right brain wants more connection. It craves complexity. The right brain will often rest in fractal complexity, such as a tree or a mountain or a flock of birds flying or a rich work of art or clouds. The right brain sees a fully 3D world, with a full understanding of the concept of depth in all it perceives.
The right brain recognizes changes in the visual field. In the periphery it will notice small changes in luminosity and shift the eyes (and head if necessary) to more fully see it. It can recognize many things. Some things like snake-shapes and spider shapes may come from our genes. Others are learned (via the right-left-right paradigm). For some of those things the right brain initiates instinctive or learned responses. For others, it alerts the left brain. All the while soaking in the world and seeking more.
The most complex thing the right brain recognizes is the face. The right brain tries to find faces everywhere: human faces, pet faces, other animal faces, even insect faces, all turned at different angles. It not only finds them, it identifies them from the subtlest of cues. While it is identifying a face, it is also reading subtle (and not so subtle) emotional cues, so we can look at a face and instantly recognize what that being is feeling (though not flawlessly.) Then we can tell where the face's eyes are looking in the 3D space that surrounds us. Then we can read the dynamics of a face (and its body) to predict its behavior (again, not flawlessly), all before the name of the face owner pops into our conscious thought.
The same goes for all sensory fields. The healthy right brain wants to touch and be touched. It hears the environment. It pays attention to the brain's mental states. These things go on whether you are attending to them or not, and the right brain will try to get the attention of your consciousness when something interesting or important happens.
The primary effect of this robust connection with sensory input and attraction to complexity is that the right brain perceives connections with everything it surveys. It doesn't 'think' about the connections in any conventional sense: it resides in those connections. It a very real sense it is those connections. The right brain doesn't perceive boundaries between itself and the world of its senses. It doesn't even perceive boundaries between the various things it perceives. It perceives patterns in a unified landscape of sensual data.
The right brain lives in time as much as it lives in the 3D space of its senses. The right brain experiences time as a continuous flow of changing sensory input, reasonably calibrated with real time. Events not only happen in order, but intermediate events are experienced and remembered for just about every two events.
The right brains spends its time and energy resting in the continuity of the connections it has with the greater world as the world changes over time. It reports to the left brain many interesting patterns it recognizes, and it does this tirelessly, providing a reliable context for the more frenetic activity of the left brain.