I know I need to aerate the small backyard (50 x 100 feet), but ...
I see this ALL the time. I maintain firmly that you don't need to aerate. What you need to do is water properly. Chances are your soil is in pretty good condition, especially if you have had full weed cover over it. If you have had bare spots, spread poisonous chemicals, or allowed a flood to kill the beneficial fungi in spots, then you might be a candidate to take extra measures, but you will very rarely find me suggesting mechanical aeration. Water works very well all by itself. Here's how.
Before I get into how, let me ask this question. After a deep penetrating rain, is your turf soft to walk on? If so, you do not need to aerate. If it is still very firm underfoot, then you need to do something (not aerate). Okay, not here's how...
If you soil is always hard underfoot no matter how much rain you've had, then I suggest you get at least 100 feet of black soaker hose as your only tool. Soaker hose can be tricky to get even dispersal. You have to run it at the same elevation. If you have turf on a hill, the soaker has to run across the hill not up and down. I run it starting at the top of the hill going back and forth, 18 inches away on each pass, across the slope until I run out of hose. Then at the faucet, turn it on to a trickle strength and go away for 3 weeks. There has to be water coming out but just don't over do it. After 3 weeks, move the hose down the slope of the hill about 18 inches.
What this does is allow the soil microbes under the hose to get a good drink of water as well as form a capillary pathway down deep into the soil. It could be your soil was too hard and you got runoff. Well if you have the hose on a trickle, it can't run off and has to soak in. When the microbes get a good drink, they will wake up and start to multiply and grow. It is the microbes that give your soil tilth and aerate it for you. You can punch thousands of holes in your soil with a mechanical aerator but the microbes can punch billions if not trillions of holes in the soil. Once they do that, your soil will be nearly too soft to walk on after a rain, but the surprising thing is that it will become very firm a week to 10 days later. In other words it becomes a sponge. Once you wet it it's soft. When dry, it is hard.
In order to keep your soil tilthy you have to keep a 100% grass coverage or mulch/compost the bare spots. Secondly you need to adjust your watering program so that you do not water frequently for brief periods. The worst watering program is to water twice a day for 5 minutes. The best ones actually rely only on rain but you will probably need to water weekly for an hour or so during the severe heat of summer. And then the last thing you can do to help yourself is to keep your grass dense. For most grasses that means tall. Bermuda, centipede, and bentgrasses are different and get the most dense when mowed under 1 inch high.
All this presumes you are on a full organic program, not a bridge program or one of those programs where you sometimes use chemical poisons when nothing else is convenient.