After doing some work in my garden there are areas of the grass with dirt on it. What can I do to remove it effectively?
Pick up what you can with a dustpan and stiff brush, or other means, then just use a stiffish broom (not too stiff) to brush it in order to spread it out, till it more or less disappears. If this is caused by where you've dug plants up and put them on the grass, next time, use a tarp or sheet of plastic laid on top of the grass, then fold over, pick it up and tip the soil on open ground. Much easier that way...
The tarp thing works so very well especially to collect yard debris. I have to tell you this is no big deal for your lawn. I charged money to 'top dress' or dump soil on the lawns. We used big wide not too soft of brooms to knock piles of soil we dumped on the lawn. Then we'd reseed and roll. Grass is pretty tough stuff.
After aerating leave all the plugs of soil on your lawn without messing with them at all. Mole hills, just knock them down with a broom and be glad they are top dressing, eating grubs and aerating your lawn bed, for free! I am much more worried about the health of your lawn.
You are mowing either too short or this is warm season grass. The color shows that fertilizer is needed and soon. Don't use a fast, synthetic fertilizer...go find a good organic (extended release) fertilizer. My favorite (and I used this commercially without charging my clients more) was Dr. Earth's Lawn Fertilizer. More expensive but so very worth the extra bucks. Doesn't have to be Dr. Earth...but look at the ingredients and shop around a bit! These new type of organic fertilizers are fantastic. I am not an 'organic or natural' label lover at all but these slow release fertilizers blew me away with the results. This one has bacteria included that is necessary to decompose thatch. Far slower results than Scott's or Ortho's intense and fast fertilizers but the slower, longer acting fertilizers I think are healthier for plants. The other upside is that it lasts longer. I only had to use it 2 or 3 times per year versus 4 times per year.
I'd like to know where you live and a picture of your entire lawn. This picture is showing too much space between grass plants. Which means lots and lots of room for weed seeds to germinate. Very stressed grasses. When did you aerate last? Have you used any other chemicals such as lime, moss killer (sulfur), weed killer? And if these are cool season grasses please do not mow any shorter than 3".
Everyone loves to emulate a golf course. Golf courses use shallow rooted grass that are usually a 'weed' grass like Poa annua. They choose this species mix so the grass is able to be mowed short, have little debris, yet needs constant fertilizer and constant daily watering. Sadly, the golf course grasses ruin all the lawns of the homes built around the edges of the golf course. Birds eat the grass seed of the Poa annua and poop it out on the dark green lawns of the homes. The golf course grasses are lighter in color and shallow rooted. Even healthy lawns of the residences that were mowed no shorter than 3", fertilized correctly, watered deeply to train the roots to be deep and drought tolerant easily discouraging any shallow rooted grasses to germinate...still got this lighter grass of the golf course to take hold. Looked awful. My clients gave up and I removed the lawns to install gravel walks, patios, plant beds instead.
Mow no shorter than 3"...not 2 1/2 but 3". Water one inch per week and train your lawn to be able to enjoy one deep watering per week. Lots of other lawn prescriptions on this site in more detail you should read. You'll only be buzzing the tops of the grasses each week with little debris (I believe in bagging and using the debris on plant beds thinly, I've not found a true mulching mower). Send more information if you are able!