Streams do most of their erosion, transportation of sediments, and deposition of sediments during?

The faster the flow of water, the more sediment will be
stirred up from the bottom of the stream.  When more sediment is stirred up more of it
is transported and deposited downstream.  The greater the volume of water the more
sediment from the banks of the stream breaks free and is transported down stream. 
Erosion also exerts its force faster when the speed of the water flowing down stream
increases.  That said, any conditions which increase the volume or velocity of the water
in the stream will increase the predominance of erosion and sediment transportation. 
Ideal conditions for this are rain and melting snow; so during any rainy season or in
the spring when the snow melts into the stream erosion and sediment transportation will
be at its peak.