Ground vias and completing a board
You won't need vias to connect your IC grounds to a ground plane.  You are using through hole parts.  The holes are plated, and basically they are oversized vias.  Any pin that connects to ground would automatically be connected to the ground plane - provided that you manage to define the copper area as a ground plane.

As for creating a ground plane in fritzing, that would seem to be fairly easy.

From the fritzing forums:


  To create the ground fill you just need to right click on a connector or pad that is part of your V- (GND) net and select set as ground fill seed. If you have more than one circuit with separate V- (GND) nets then you will need to set at least one seed for each net. After that make sure both layers are set as clickable (or which ever layer you want the ground fill on if you only want it on one) then go to the Routing menu > Ground Fill > Ground Fill (Top and Bottom).




Mmmpf.  Forgot the part about the board layout.

You've got traces front and back on that board.  That can chop your ground plane into little, unconnected pieces.  Try to route everything on the top side, and only use the bottom for short jumpers to get around crossovers on the top.

The layout itself looks reasonable for what you are doing - I don't see anything that looks particularly tricky.

What I do notice, though, is that you have no decoupling capacitors for your ICs.  

You might get away without them, but maybe not.  

You should add small capacitors (typically 100nF) at the power pins to each IC.  I mean right at the IC.  Shortest leads possible, shortest traces possible.  Each time a digital IC switches state, it draws a really short pulse of current.  If they all do that at once, then it can cause the voltage to each IC to drop - and then the ICs get stupid ideas like freezing or giving random outputs.  The little capacitors catch the current spikes and keep the voltage steady.