A good start for learning how to lightning components work would be through Trailheads Lightning Components Basics
You should have (or should) declare an attribute for your date as specified in the doc I linked for
<aura:attribute name="attrName" type="Date" default="" />
and in your controller set the value accordingly:
you will need to reference it in your lightning:input component as you had done originaly using
v.dataInizio however, you seem to have changed the attribute name to value for some reason.
here is a little screenshot of both components compared :
notice that the date formatting has to be formatted as YEAR-MM-DD for lightning:input to accept the value or it will default to:
and yes, if you use an aura:id, you can fetch your component and set a value to it without using attributes.