Direction of Current Flow in the circuit?
Yes, D1 and D3 conduct at the same time, as do D4 and D2.

This configuration of diodes is so common it has a name, which is full wave bridge.  You often see it drawn in a diamond shape as in your schematic, but I like to draw it differently to avoid exactly the confusion you are having.

Here is a better way to visualize a full wave bridge:



Imagine these are perfect diodes for now.  Regardless of the polarity of the AC signal, the more positive AC line will make one of the top diodes conduct, and the more negative one of the bottom.  Hopefully you can see how this effectively takes the absolute value of that AC input voltage.

Real diodes have forward voltage drops.  If we figure 700 mV for these ordinary silicon diodes, then the AC signal has to have a magnitude of 1.4 V before anything conducts to the output.