I was recently given a Squire Strat from one of my friends with a VOX AD15VT amp. While the amp has tons of effects built in, I was looking to start getting pedals in order to have more control over the effects and be able to chain them together.

I guess I was just wondering what would be some good starter pedals that I should pick up?

I want to be able to stuff by bands like Brand New, Portugal. The Man, Taking Back Sunday, and Coheed and Cambria. For Portugal. The Man, I know I should probably get a delay and reverb, but I'm not sure what else I should be looking for.

Should I even be bothering with pedals right now, or just look at getting a better amp?

I just don't know where to even start with looking at pedals. Electric guitars are still kind of a new world to me, as I'm learning about pickups and amps still.
If I was in your position, I'd not bother with pedals just yet, though it can't hurt to try out a couple at stores if you get the chance, just to see what you like and what not.

As far as the Vox goes, just get the VFS2 footswitch for it (though probably, any two-button footswitch will work). The footswitch allows you to switch back and forth between the two channel-presets on the amp with one button and can be used to switch the effects-section on or off with the other. That's quite a bit of control for comparatively little money. I wouldn't recommend spending much cash on pedals if you don't really know what you want.