WRT3200ACM backup/restore

Just got this router today. I already have 18.06 running on it. The partition layout on it is not what I would normally see. Looks like:

root@OpenWrt:/# cat /proc/mtd
dev: size erasesize name
mtd0: 00200000 00020000 "u-boot"
mtd1: 00020000 00020000 "u_env"
mtd2: 00040000 00020000 "s_env"
mtd3: 00040000 00020000 "devinfo"
mtd4: 001e0000 00020000 "sysdiag"
mtd5: 05000000 00020000 "kernel1"
mtd6: 04a00000 00020000 "ubi"
mtd7: 05000000 00020000 "kernel2"
mtd8: 04a00000 00020000 "rootfs2"
mtd9: 05600000 00020000 "syscfg"
mtd10: 005c0000 00020000 "unused_area"

Typically, 'firmware' is in most layouts. So, if I were to backup each partition as:
dd if=/dev/mtd0 of=/tmp/[partition_name].bin

Which one of these partitions would I use to restore my settings?

If you read the wiki material on wrt3200acm (and wrt1900acX series), you can see that there is a dual firmware structure. You are currently using kernel1+ubi, but after the next flash you will be using kernel2+rootfs2.

Best option is to backup the Openwrt settings as a .tar.gz archive using the sysupgrade backup tool (or via LuCI). Trying to restore whole ubi partitions is not that practical.

Thank you for that information. I've never had a router that had dual firmware, so that's completely foreign to me.

Check the wiki and discussion. The dual firmware works as a fallback is something goes wrong. You always flash the other partition and then switch to using that. After three consecutive failed boots, the router automatically falls back to the old firmware. You can also manually switch between firmwares. See. E.g. advanced reboot app for Luci.