If I want to say "Hold on a second, I'm gonna go grab a glass of water", can I use the verb pakken? I think I've only ever heard "Ik ga een glasje water halen", but I'm not sure if I heard "Ik ga een glasje water pakken". Are they both correct?
The two sentences are correct (but glaasje, not glasje). But the English sentence is rather informal, I think, and in Holland, one would expect to hear something like "Wacht even (of: moment), ik pak (of: haal) even een glaasje water".
It's preferable to say "Ik ga een glaasje water halen" , this is why you probably never heard this phrase with the verb "pakken".
And like others said before, if you want to translate the "hold on a second", you can say "wacht even" or "een minuutje (aub)" or "een secondje (aub)".