I'd like to know how to make a sort of cartoony trail behind an object as it moves around, somewhat akin to the trails that often come off of wings of airplanes. I'd like it if it could get thinner towards the end, as well. Could somebody please help me with this?
You can add moving trails by animating the Start and End settings of a Curve objects generated to follow the path of the element (eg, vertex) where you want the trail - similar to How to add trailing light effect to finger tips
The linked answer uses particles generated from vertices to trace out the path of the motion and this is converted to a curve via a Python script. However, to make this easier to run I have converted the script to an add-on (ParticlesToPath) that can be downloaded from here.
Simply install the add-on (in User Preferences/Add-ons, select Install Add-on from File and point it at the downloaded .ZIP, then click the checkbox to enable it.
Add a particle system to your object to emit 1 particle per frame from the selected vertex. Ensure the particles are emitted with no velocity and reduce Gravity and All Field Weights in the particle properties panel to zero (so the particles don't move).
Press Space (or F3 for Blender 2.8+) and type 'Particle'. Select the Particle To Path operator.
On the popup properties window ensure you enter the index of the particle system (with '1' being the first) and optionally adjust the default bevel/curve properties.
Click 'OK' and you should now have a curve that follows the motion of that vertex. Animate the Start and End bevel properties to generate a moving trail and set taper, etc. as detailed in the linked answer.
For a cartoonish effect trail create a dummy object for your trail smoke, in this case I used an icosphere.
Create a particle emitter attached to you moving object, in this case I used a simple plane. Animate it somehow so it moves about your scene.
Add a new particle system to your plane. Make sure it emits from faces, physics as Newtonian, turn on Rotation and set Angular Velocity to Random with a value of 2 and turn on Dynamic.
Under Render set to Object and pick your icosphere (or chosen particle object) adjust size and set a randomness factor if desired.
Under Fields Weights turn off or seriously reduce Gravity Influence so they don't fall over time.
Now under the textures panel add a new texture. From the texture settings tab adjust it's properties.
It should be a Blend type texture. Turn on the Ramp option from the COlors panel, and invert it so it starts white on the left and progresses to black on the right. Adjust as desired.
Under Mapping set it to Strand/Particle and make it influence only Size in the Influence panel