I have recently downloaded Fusion and have seen some preloaded uses of fire and smoke in reference to particle systems. However, no matter how much I try recreating something like that, it always looks fake as either the particles is too thick or it simply doesn't flow or doesn't look realistic. This applies to other concepts as well such as snow, fireworks, etc. What makes the particle systems look so natural in those cases? How can I achieve a natural more realistic look from them?
Fusion doesn't have fluid dynamics simulation built in. To get a realistic particle flow of a natural system, you'll need to use software specifically built for that task, import the results into Fusion, then use Fusion for compositing the smoke by adjusting the way each particle appears on screen and how they blend with the other scene elements. SideFX Houdini with pyro, or Cinema 4D with X-particles are two pieces of software which might be better suited to fluid particle simulation.
Most people don't think of fire or smoke as fluid because they're gaseous and aren't wet. However, in the context of particle VFX, fire and smoke are fluids and require decent fluid dynamic simulation to appear natural.
Jason isn't incorrect, however it is absolutely possible to achieve a realistic effect without fluids. I've done major FX in 3DS Max for years and yet many times I've been able to solve some pretty cool simulations in Trapcode Particular alone. The key is usually to get the turbulence right, which is what gives it that "fluid" effect. (also called voritcity, or wind in some applications). I don't know fusion well but one trick I've used many times is to combine 2 different turbulence's - one which is large and gives you the random shape, and one which is smaller and gives you detail. If you think of a campfire you can see big loopy curls of fire, which often fork off into 3-4 small jagged edges so to speak.
Another good tip is to find footage of your desired effect and study it's motion. Finding a good fire/snow/smoke reference online and watching it over and over you will learn things you never knew about how it moves and reacts.
And lastly as a compositor I HIGHLY suggest getting some real elements to layer into your effect any way you can. For fire you're going to want to find some sparks to put over the flames, etc. Don't underestimate this one! It's easy to find free footage online (mitchmartinez.com) and layering a few of these on top of your simulation is how the pros do it.
All that said check out the new Trapcode Particular. It has a "fluid effects" system which is kind of blowing my mind. For someone relatively new to CG effects it is a dream come true. It took me weeks to fully understand FumeFX (which has been an industry standard for fire/smoke) to where I could use it efficiently, so I don't really suggest going down that route unless you want to make it your job. Particular however you can pick up in a few hours and it is relatively cheap. Only downside is you'll need to also buy/learn After Effects.