I think to do a drywall ceiling is the answer but you will bneed to realize you will have spaces with lower ceilings to accommodate the HVAC . So a good design will let you do the drywall and keep the low ceiling in spaces where it is not a big deal like the laundry space or a kids play space .Some of the piping can be moved to me it looks like the basement was never planned to be finished so talk to a plumber and an HVAC pro to see what can be done .

How much ceiling height do you have to work with? Bulkheads to conceal ductwork even if they're low aren't necessarily a problem if they aren't in a traffic pattern. If you have enough height to work with, covering it all up with drywall is the treatment that returns the most value upon resale. You already have pictures of everything. If at some point you need to cut out a piece to get to something it is good to know where everything is. But in general ductwork doesn't need to be accessed once covered. Sometimes pipes need to be, but even that might be only every 30 years. Think about what you want to use the space for - maybe you don't need to finish all of it. We have/had a partially finished basement in our last house and our current house. That is actually my preference. It's nice to still have unfinished space for storage. Plan for that near the furnace and hot water heater while making sure you don't crowd them. In our current house the previous owners finished part of the basement and have a closet under half of the big duct (trunk line) so it's not visible. I thought that was clever. Instead of having multiple bulkheads to hide things individually, try to group them together to have a smoother appearance of the ceiling. or create a rational pattern with faux bulkheads. In the first picture I'm not convinced they had to drop all of that - part of it may be just to make it look intentional.

Spend a lot of time thinking about how best to lay out the space. A contractor may not be very much help. Sometimes they just want to throw up drywall wherever you tell them to. After all, they're not going to be living there afterwards and a lot of them don't have any sense of artistry. If you don't already have vents in the ducts down there think about that, and pull any necessary electric and/or plumbing before closing it up. Finally, check your local code to see if there are rules around ceiling heights in basements before you start work.

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Are your intentions to DIY or hire a contractor? In either case you will need to anticipate usage, know some basic electrical and utility needs and provide initial space planning suggestions.

Ours was a DIY. After measuring, then marking all permanent features on graph paper, we placed masking tape on the floor to position walls/features to be added. Light locations were marked too.

For our ceiling, 2x4’s were installed across the joists to clear existing electric, gas and water. So 2” height has been lost. Ducting was encased similarly by dropping a pony structure and bridging across.

The plan below was faithfully followed. It shows measurements and locations. Hopefully provides some starting point help to get you going. Easier now with design software avaiable.