Maintenance Checklist for a "New to Me" 6.0L Diesel:
Stage 1: Full maintenance workup
5W40 synthetic oil, with a motorcraft/racor filter and a factory oil filter cap <<< cannot stress enough checking the oil cap immediately, incorrect cap and filter combo can damage your high pressure oil system
Fuel filters (Racor or Motorcraft brand)
Coolant exchange (also install high idle mod, recommend a Cat EC-1 rated ELC coolant)
Trans fluid drain/fill and external filter
Drain water separator
Replace the air filter (Donaldson or Motorcraft filter with blue media)
Clean EGR valve
Load test batteries, replace if fail
If it's not there already, find a way to put the stock air filter back in it

Getting the baseline taken care of will be important, to get started on 5k mile oil changes, 10k mile fuel filters, and 30k trans fluid/filters. The air filter doesn't have to be replaced unless the vacuum indicator tells you it needs it, but changing it now gives you a chance to put a new, correct filter in that can last you 50k+ miles.

Stage 2: Common Updates/ Maintenance Mods
Coolant Filter
Blue Spring fuel pressure regulator
Gauges - absolutely a must have (Info Here)
Replace degas bottle cap
Updated brass drain plug in HFCM
FICM voltage testing, repair if necessary

Gauges are the most important thing here, because they'll tell you what parts of the next stage you need to take care of. There are a bunch of threads in the 6.0L forum that'll tell you what to watch, what to look for, and what problems are caused by certain readings.

Stage 3: Common Engine Failures (if needed)
STC Fitting (2005+ engines)
Standpipes and dummy plugs (2005+ engines)
Oil cooler replacement
Bullet Proof EGR Cooler replacement
HPO reservoir screen
Replace CAC boots if oil soaked
Replace plastic cold side CAC tube with metal if cracked
Turbo overboost/underboost from rusted unison ring
Dusted turbo vanes from using an aftermarket air filter
Upgraded Alternator and wiring
FICM tuning, with stock PCM rollback

Do what you have to do here, some people just do what's broken and some people replace it all at once. Depends on your budget, skills, and how long the truck can be down.

Stage 4: Power Adding/Bling
SCT Tuner
Studs, stock gaskets, leveled heads
Injector/turbo upgrades
Leveling/lift kit/tires