Points of interest and amenities are probably the next most useful. Things like street lighting and sidewalks quite helpful for pedestrians. It's more important that you are still enjoying mapping. If there is something that you know is useful but you find tedious it's ok to skip it and either leave it for someone else or come back to it later
As it sounds like you're on Android, I'd recommend @streetcomplete for easily adding detail while on the go.
@pootriarch
I generally consider it poor form to remove good data even if it isn't strictly necessary.
The necessity of the addr:city tag depends on the area. In some places it's strictly according to a boundary that exists in OSM. In others the boundary data is either missing or doesn't always align with the "postal town" according to the mail carrier so it's good to include it.
@openstreetmap