Just do 'Factory Reset'.
Well, this depends upon the recovery you're using. If you're using TWRP (assuming you've v2.8.4.x or above), choosing Factory Reset would remove everything under /data (excluding /data/media which serves as emulated storage available through /sdcard as internal SD card), would wipe cache partition, and since davik cache gets stored either in cache or data partition, it would already be wiped out.
If you're using stock Android recovery (<3e>) and choose Factory Reset, then it would format the data partition and cache partition. Davik and internal SD card would also be automatically taken care of by formatting.
If you're using Cyanogen recovery or any other recovery, then I can't comment. I can only suggest switching to TWRP for better control and features.
Do 'Factory Reset' at least 3 times.
Do 'Factory Reset' and wipe dalvik cache.
You only do it one time and that's all you need to do. As I said, a factory reset automatically wipes cache partition so you need not to explicitly wipe it. TWRP notes:
[You] need to [factory reset] and you only need to do it once, not three times as some people would lead you to believe.
As for,
I'm interested in erasing all traces of my personal data from the device
Wiping the partition wouldn't be enough if you're talking about erasing all traces of personal data. There are already Q&A discussed about that on this site, so I'm referring them to you:
Note that if the phone would remain with you after a factory reset/ full data+cache wipe then it is not worth the time engaging in addressing "traces of personal data" since resurrecting deleted data requires physical access to your device.
Are there any options I should steer clear from? I would like the phone to boot to a fresh CyanogenMod after clearing all data.
Yes, please stay away from anything other than /data, /data/media and /cache. System partition doesn't contain personal data and when you would flash your ROM the partition would automatically be flashed because modern ROMs (such as CyanogenMod) do that. Modem partition is mounted at /firmware so it is not worth wiping it out because that may only cause trouble for you.
Related: Won't placing a cuostm ROM file before factory reset remove it from the phone?