Choosing threshold's is always tricky. You want to bring in enough data to understand the correlation between your genes of interest, but not so much data that you get a hairball. My suggestion would be to start by doing things interactively until you get a useful result. So, bring in your hairball, then go to the "Filter" tab in Cytoscape. Create a Column Filter (the + button) for Edge:weight. This should bring up a slider and allow you to interactively select edges with lower weights. Start by setting it to 0.5 as a starting point. Now, there are two possible things you could do next:
Delete the selected edges, and then relayout the network. If that looks better, you can repeat this cycle. If you've gone too far and you wind up with a number of singletons, then hit "undo" to undo the layout, then "undo" again to undo the delete. Then you can select a lower threshold.
You could also invert the selection (Select->Edges->Invert Edge Selection, Select->Nodes->Select all nodes) and then make a new network from the selected nodes, selected edges.
I would actually recommend #1 to get started and only go to #2 if you are trying to really compare between two values that both look good.