Just a note: there is no kernel 3.13 in that repo.
It contains 3.16.1 at the moment… 
when I restart, I get
error: you need to load the kernel first
Apparently the grub menu entry is wrong.
Which grub are you using? grub1 or grub2?
If in doubt, post a picture.
and then I can only boot into recovery mode.
You should be able to do a normal boot with the older kernel as well.
also, when I do,
uname -r
shows kernel 3.11.10-21-desktop
Well, you apparently selected to boot kernel 3.11.10-21-desktop.
So did I screw up my installation entirely?
No.
You just have an incorrect boot entry in the boot menu.
As openSUSE keeps more than one kernel (and boot menu entries for all installed ones), this is no problem.
Just choose a different one. (with grub2 you have to select “Advanced Options”)
Was what I was trying to do the right way to update the kernel to 3.13?
No, because that repo only contains kernel 3.16.1. 
I’m not aware of any repo with kernel 3.13, so you would have to download and compile it yourself. But why? 3.13 is EOL since quite some time, better install a supported one like 3.12, 3.15, or 3.16.
Other than that, your steps were correct though.
But again, they will install kernel 3.16.1 (at the moment), not 3.13.
Since the boot menu wasn’t written correctly when you installed the kernel, better check that you have enough free space on /boot:
df -h /boot
Hints:
- You can install/remove specific kernel versions in YaST->Software Management, by using the “Versions” tab below the package list.
- You can edit the boot menu/boot loader settings in YaST->System->Boot Loader. If you use grub2, just calling that and pressing OK will re-create the boot menu and might fix your problem already.