There’s not much of “game” to the Duty Officer system. It’s more of a subsystem for, essentially, gathering. But there are mission chains that link together and different missions in different sectors and worlds. So learning where to go to have the best chance to get a raw material you might need, the schematic that goes with it, and the chance to make it is part of the system. You can end up making consumable ship components, Bridge Officer gear, recruiting duty and (high quality) bridge officers, getting special temporary bonuses, and other things in addition to gathering experience, bridge officer points, branch experience points, transporter credits (mainly in trading missions) and raw dilithium.
Some missions can also be dangerous. You can have green or better crew sent to sick bay for some time. White crew members can even get killed.
I like it because it keeps me engaged and gives me lots of low impact filler when I’m tooling around the galaxy. I’m usually not big on gamefaqs style cheats so there’s also an element of discovery, that’s fun, when I stumble across a new mission somewhere.
I can also pop in for a short spell, load up on missions, and then log out. Every so often I might log in just to check on mission progress as well. Some days, when I’m spending lots of time in the game, I might get into micromanaging the process and bypassing my first officer team recommendations so as to get the absolutely best crewmen assigned for a given task or, conversely, to keep the more qualified crewmen free and available for other missions that might crop up.
Also note that you can assign Duty Officers to permanent, but reassignable, positions managing space or ground tasks. These give direct bonuses or chances of a proc to certain actions. The better the quality of the crewman assigned the better the effect or chances of a proc. But assigned crewmen can’t also be used for Duty Officer missions. You have to decide if you want your very best men on station or available for deployment. I usually keep my most relevant purple and blue quality officers on station but if some very special (rare or very rare) mission crops up I may pull them, temporarily, and set them on that task.
Currently I’ve expanded my base DO crew slots to 200. I managed to fill up the first 100 slots with a mix of officers with useful specializations, races and traits for variety. I ended up wanting more slots to make room for recruiting and special characters like prisoners and colonists. At this point I usually crit about 10% of my missions, fail about 10% and critically fail only rarely. I haven’t gotten so involved, yet, that I go roaming into particular sectors looking for special missions. At most I might hit a sector where it seems likely I can grab some missions that will give me contraband which I can then use for some special security missions or in missions that develop new technologies.