I keyframed an effect and would now like to find a way to cut it up a bit, so in other words, the current effect lasts 10 seconds and I would like to somehow maybe turn off the effect after 1 second, turn it back on, turn it off another second or so later, turn it back on, etc. If this was a clip in Premiere I might use the razor tool to cut the clip up and close the gaps to achieve something like what I am envisioning. How is something like this done in AE?
That depends on the effect. What are you doing?
Here's the Premiere-style way to do it -- split the clip into sections. Remove the effect from the layers's where you don't want it.
The split clip keyboard shortcut command is ctl-shift-d (Win) or cmd-shift-d (Mac). The split takes place at the position of the timeline cursor.
In the Timeline panel, every effect has a Compositing Options property group with property with which you can keyframe/animate the opacity of the specific effect. Animate that.