Keep off the grass
Grass needs sunlight, water, nutrients and air to grow so this way you’ll get oxygen and water to the roots while releasing carbon dioxide that has accumulated. Spiking with a garden fork is one way to do it until you can get the garden tractor with a spiker/aerator on the lawn when it’s drier.
Compacted soil under the grass can lead to thatch and moss which are best treated in the spring by scarifying. Moss can be treated then with moss killer. If you don’t want the unsightly black residue to rake out, the product MO Bacter is an organic fertiliser which feeds the lawn, while causing the moss to break down and become ingested by bacteria in the fertiliser. It leaves no debris and won’t need scarifying. It’s best applied in spring into summer.