Identify where the sun's rays strike earth most directly and least directly

-- Depending on the time of the year, the sun's rays strike Earth
most directly somewhere between the Tropic of Cancer and the
Tropic of Capricorn. 

That's a belt around the Earth's "middle" called the "Tropic Zone". 
The equator is in the middle of it, the Tropic of Cancer is 23.5° North
of the equator, and the Tropic of Capricorn is 23.5° degrees South of it. 

The sun's rays can never be totally direct, straight down onto the Earth's
surface, anywhere outside this belt.

-- The sun's rays strike Earth least directly wherever, and whenever,
the sun is setting.