Which of these will work? HDMI to VGA or HDMI to USB?
You won't see video if you connect through your USB port. Only HDMI->DVI-D or HDMI->VGA will work (relying on the updated information just provided about your monitor), and DVI-D connections offer higher resolution and better quality.
The USB ports on your monitor is a convenience feature to pass USB through to peripherals (e.g., keyboard and mouse) near the screen; you have a Type B (square) port to connect to your PC, and two Type A ports for keyboard, mouse, and/or other peripherals.
Here are pictures of the popular video connectors, if you want to double-check. 
The Dell P2412H has a DVI-D interface. You can get an HDMI-to-DVI cable (Amazon US has them for ~$7).
Or you can use an HDMI-to-DVI adapter (about $6) -- effectively converting the monitor to HDMI input and then use an ordinary HDMI-HDMI cable (such as one you might use for connecting a recent games console to your TV).
I've got both kinds and use them to connect my (HDMI) laptop to my (DVI-D) Dell 2205FPW monitor.
Those cables seem remarkably cheap. More expensive cables might be better, in terms of supported resolution, etc. (up to a point, where you're just being ripped off).
The USB connector on the monitor is because the monitor has a built-in 2-port USB hub.
This is particularly useful if you have a laptop: you can leave a USB mouse and keyboard connected to the monitor, and then you only have to plug one USB cable (the monitor) into the laptop.
If you have a desktop (that doesn't get moved), this might not be as useful, but it can help to keep your desk tidy. I have my webcam plugged into my monitor hub.
One thing to be aware of: a lot of monitors (including all of the Dell monitors I've used) turn off the USB hub when you turn off the monitor, and sometimes when they go into standby (which means you can't twitch the mouse to wake up the PC).
It’s very likely the USB port(s) on your monitor are just a USB hub and cannot be used to display images. That leaves the VGA port as your only option.