Here's the first thing you do: don't try to use AE intuitively, which it sounds like you're doing.
Instead, get a good grounding in the basics which makes AE life SO much easier... and get it from the pros:
There are several ways to do this. But don't go into the menus for any of them.
There are several things you could animate to make the glow begin differently. You could split the layer and apply Glow to the second part. You could simply enable keyframes on ANY property of the Glow effect (this is probably the way to go) - which parameter you keyframe is up to you; you will need to experiment.
Having said all that, I'm pretty sure Dave's link is going to benefit you a lot more. AE is much more complicated than new users think it is.
I found the easiest way to do it was jut add an adjustment layer with the effect for part of the selection. I now realise effects don;t create keyframes as such. This is confusing but I get the difference now.
You really, really need to go through that link Dave gave you. It explains everything you're confused about.