The following are a few tips I've learned to make working with connectors in Visio a little smoother. Feel free to contribute your own in the comments.
Set the Line Jump Style to GAP
Visio's default method of depicting connectors which cross but do not intersect is to illustrate one line arcing over the other. This is great for electrical drawings and other schematics, but isn't always accommodating of network topologies, especially when one line intersects a number of other closely-spaced lines.
For a cleaner look, we change the line jump style to "gap," which renders aesthetically pleasing white space to highlight line crossings. From the Developer tab on the ribbon, select Show ShapeSheet > Page. (If you don't have the Developer tab, go to File > Options > Customize Ribbon and enable it.) The page's ShapeSheet pops up in a window consuming the bottom half of the screen. Under the Page Layout heading, double-click the LineJumpStyle key and select "2 - visLOJumpStyleGap" from the available options. Press enter to save the selection.
Turn Off Rerouting
Connectors in Visio tend to be very timid and jumpy: A connector frequently will re-route itself when a shape or another connector moves in next to it. For example, observe how the connector below automatically shifts up and to the left a bit when the router shape at bottom is dragged close to it. This behavior can be quite frustrating when trying to keep connector lines predictable and evenly spaced.
Fortunately, we can disable this behavior. Select one or more connectors, then open the Behavior menu from the Developer ribbon and select the Connector tab. Set the reroute parameter to "Never." The connector(s) will now stay firmly put regardless of how many shapes you place on or around them.
Unfortunately, I have not found a way to automatically apply this to all connectors in a drawing; it must be performed manually once the connector has been drawn.
Change the Default Connector Style
One complaint I have with Visio is that there's no simple way to change the default connector style; for example, to increase the default weight of 1/4 pt to something we can see more clearly. The habit I've held for years is to draw a number of connectors, manually select them all, and adjust the weight of them all at once using the ribbon tools, and repeat as I go. However, there is a convenient way to modify the default connector style using themes.
From the Design tab on the ribbon, select Effects > Create New Theme Effects... and name your new theme something meaningful (e.g. "Thick Connectors"). Select the Connector tab (not the line tab, as regular lines are themed independently of connectors) and the desired weight. Save your new theme and it will be automatically applied to the current drawing.
Now when you draw new connectors, they will be created with the weight you specified in the theme settings. Consider tweaking other parameters of your custom theme as well in order to save time on other common reformatting actions.