A good problem statement is characterised by a good understanding of who experiences the problem (your customer or user), what their objectives are (what task do they need to get done) and what constraints they are under.
Additionally any insights around painpoints, emotions, barriers or other that you are aware of will add value and context to the following statement:
It is hard for… [this customer group], to do… [this valuable task], with… [these constraints, insights].
Google has lots of obvious and less obvious answers to your questions http://google.com
When talking to people, the primary objectives are to unearth the core needs and painpoints people (your future customers) experience. These can further be categorised in functional and emotional aspects. Functional elements can be referred to as "tasks" or "jobs to be done", do check out the below resources on JTBD. Emotions are really important as they will drive future behaviours and motivations to use and buy your products and services.
Technique 1 - Jobs to be Done | The Innovator's Toolkit
Jobs To Be Done (JTBD) are a great concept to understand the core functional needs and goals of your users.