With the heavy rains this monsoon, the dirt driveway and yard at my new home has gotten pretty eroded. Any tips on how to fill or level the erosion? Special tools smaller than a Bobcat? Is there anything meaningful I can do to reduce such erosion in the future (other than installing some man-made surface) or is this just a fact of life for me now?

Rent a small tractor with a box blade.

Start where the soil/gravel washed to and drag it towards the top.  The box blade, for the most part, will drop the eroded material back in the low areas.

This is typical for AZ when it rains.  Larger rock in the areas that the water ran may prevent some erosion in the future.

Agree with the above. Pretty sure you said in another post you are out in my area now, so a recommendation is "All American Rentals" on Apache TRL. They will deliver a small John deere with a bucket and box blade for a reasonable rate. We also have plenty of  gravel yards that deliver a load of fill dirt or gravel.

Just remember,  water runs down hill. It seems like "well, duh!', but frankly that is what you need to plan around.  Driveways can be built up a bit and "crowned" in the center so that water runs to the sides, and then flows down hill in established ditches. No need to apply pavement. Yards can be planned to take advantage of this, creating swales that capture the water for plants like boganvillia, that likes infrequent but deep watering.

The good thing is you know where the water wants to flow.  You will have to give the water somewhere to go, otherwise it will do the same thing at the next large rainfall.  Without doing some kind of paver or concrete, it's going to repeat this again.  Some very simple shallow trenches filled with rip rap will help stop this, if you route it to the sides of the driveway.  This will create a man made 'wash' to help steer the water.  The rip rap (larger 3-8" fractured decomposed granite), will help stabilize the trenches to keep them intact when the water flows through it.  Otherwise they will end up getting destroyed with any significant rain.  If you have boulders on the property, you can use those as well to stabilize the 'wash' and make it more visually appealing.

Fill in the current grooves with 1/4" minus decomposed granite (Madison Gold will match closely with what you got).  You don't need the stabilizer as it will naturally firm up after you hose it down a few times.  It'll provide a firm surface that won't immediately turn to mud in the next rain, although with much rain, it will fail when the dirt under turns to mud when saturated with rain water.  Or use the fill from the trenches you dig up as you will need to put the dirt somewhere.

You can rent a Bobcat or go at it with a shovel and wheelbarrow.  Think of it as a fitness program.  Between eating well and landscaping my yard, I was able to cut down on my weight and felt proud that I did it instead of hiring a crew for $700/day.

Thanks all - I'm adding the term "rip rap" to my day-to-day vocabulary

Say Rip Rap Rock 10 times fast.

Richard, after you restore the dirt, it may be expensive, but look into "soil cement".

Basically a lime solution commercially applied that locks everything into place.
We use it at work on all of our berms and run-out areas.
It works really well.

Oh yeah, and rip-rap.

Redirect and slow it down is the name of the game. The more it channelizes, the more it will erode it. Try and contour it to get it off the edges, and then try to create a series of steps with your riprap to keep it from getting good flow.

Waddles can also be used to disrupt the flow, but they're ugly, and have to be replaced periodically. Good for protecting a fresh surface while you pack it though.

Long term, you can look into geotextile fabric to underlay the surface, that's going to be an expensive solution.