What important items are missing from GDP?

Lots of things.
First, it excludes things that affect our standard of living, but that are included as part of measured GDP: crime rates, happiness, the quality of the environment, the sense of community, political freedom.
Second, it excludes things that are quite explicitly produced but that do not enter into a commercial transaction. These include providing your own children with home health care and education; mowing your lawn, etc. Note the arbitrariness of this exclusion: if I mow my lawn it is not in GDP, but if I pay someone else to mow my lawn it is in GDP (at least theoretically). Activities such as this are collectively termed household production, and their exclusion can lead to some serious errors in evaluating a country's standard of living.
Third, GDP excludes the loss of assets due, for example, to natural disasters.
Note the common theme here: GDP fails to include things that do not go through a commercial process by which an explicit price is revealed. If we don't observe a price, it is very hard for a statistician to know what value to attribute to the good (and this value is needed to calculate GDP).