What variables in brewing have the biggest effect on taste? I've been brewing my coffee with an AeroPress for a couple of years now and I've become aware of the huge number of variables available to me in making my morning Java.  Assuming the same bean and roast level, I see 


amount of ground coffee, 
grind coarsness,
temperature of the water,
steep time, and 
using the AeroPress right side up or upside down, 


as the variables I can play with to vary the taste.  Some of these variables may offset each other as well.  For example would steeping coarser ground coffee result in the same taste as less time steeping a finer grind?  Or how about coarser ground coffee in hotter water v. finer ground at lower temperature?  

I'm searching for the key variables in my control to play with to maximize the effect on taste.
A few other factors to consider:


Freshly ground beans.
How you store your coffee beans. You basically want to keep them reasonably cool and away from oxygen, sunlight, moisture and variations in temperature (which is why you don't keep them in the freezer except for long term storage). Generally in the pantry in an opaque, air-tight container. I one-way valve is recommended as recently roasted beans will emit CO2. I bought a stack of 250g resealable foil bags with the one-way valve from a local roaster and would use each bag a few times. These days I use AirScape canisters.
The water you use. If you wouldn't enjoy drinking your tap water, don't use it to brew coffee. Use the same water you use for drinking water.
The amount of water you put in the Aeropress vs. the amount of water and/or milk you add afterwards (if any).
How vigorously you stir with the paddle. Generally you want to stir just enough to ensure all the grounds are thoroughly wetted, and that there are no air pockets. Stirring too much can over-extract which will increase the bitterness.
When pressing into the cup, do you only press until you hear the air hissing through, or do you keep going and compress the puck, squeezing out every bit of water you can?[1] The latter can over-extract and increase bitterness.
Using paper filters vs. a metal disc filter. Also, some people believe pre-rinsing a paper filter makes a difference.


Obviously there are a lot of variables, and people can become amazingly pedantic. Fun if you don't mind a bit of experimentation.

With regards to water temperature, the general wisdom is that you want the slurry of coffee and water to brew at around 95&deg;C, which is just a bit under boiling for optimal extraction. This video gives some pretty good advice on this. Pre-heating the Aeropress would probably make the biggest difference here[2]. I usually just wait until the water in the kettle stops bubbling before pouring. Apparently it is possible to burn the grounds if you hit them with water that is boiling.

The other bugbear I have is with retention in coffee grinders. Retention is when grounds are left up in the works of the grinder, and often the grounds can come out very staticy and cling to everything and make a mess. This grosses me out as it means old grounds from previous grinding will be dropped in with the fresh grind. Weighing the beans before and after can show a variation of half a gram or more. Urk.[3] However I've found that sprinkling a couple of drops of water on the beans before grinding almost completely eliminates retention for me.[4]

[1] There's a term for over-squeezing the grounds that escapes me at the moment.

[2] Having lived a lot of my life in drought areas, using and then discarding the water to pre-heat my Aeropress or French press is something I find surprisingly hard to do.

[3] I did mention people can become amazingly pedantic, didn't I?

[4] Known as the Ross Droplet Technique