Are you starting to a bootable installer for El Capitan?
By that, I mean that you have downloaded the El Capitan installer, then used THAT to create a bootable partition?
Ideal is to use a USB flash drive for that. You can use terminal steps to make that external installer, or use one of the good apps that makes the process easy, such as DiskMakerX, Boot Buddy, or Install Disk Creator.
Any of those apps make the task easy.
And, when you boot to your installer, you will get that same set of choices. And --- you should choose "Reinstall OS X"
I have done this process literally dozens of times, and the OS X reinstall should be the correct choice for you to use.
If you still have problems connecting to internet (could still happen), then I suggest that you try connecting to your router directly, with an ethernet cable.