if you don't have drawings of the area, then take some pictures standing at each wall and looking into the room will give us a better sense of how big the area is and what it has, how many windows, support beams, etc
As mentioned you can close in the furnace area with the laundry area and put in a toilet, all in one room if your not after a fully decorated washroom, this can save you money, heck you don't even have to stud the walls and or drywall them, even the ceiling can be left as is, but do put in an exhaust fan.....
Then just stud all the other walls, close up the stairs or bring it out a bit and have under stair storage, put in a door and you can keep the mess out of sight
For the floor use that DriCore Product -
https://www.homedepot.ca/en/home/p.2-ft-x-2-ft-engineered-subfloor-panel-system.1000112986.html
it's great stuff, will make the floor feel warm and you can tile on it, or laminate, or carpet, and if any water or moisture is present it will stay below and hopefully dry out, I used this on my last house, put in laminate and was able to walk barefoot...
unless you need a specific room, ie bedroom, gun room, etc, I would just keep it open concept, and use furniture to divide the area up.
ceiling, well drywall is easy and fast, but if you want access then a drop ceiling is the way to go
pot lights for lighting, get the contractor packs, cheaper and you can install, just get an electrician to verify your work or have one do the wiring, placement you can do
the posts you can cover with MDF latter and some trim at the top and bottom to make them look a bit better, so they are not just these square posts, but they are anyways...lol
you might want to look at insulating the ceiling as well, keeps the noise down for those upstairs, and do a proper insulation job on the walls, that builder crap that is done, get rid of it and do it right...
you can use 2x4 lumber for studding, but you have to watch how each boar is installed, crown out, or use metal studs like I did, and you don't worry about that, upper channel, lower channel and put in your studs 16 on center and your good to go, only down side is you can't nail into it to hang something....but use that removable sticky hangers and your good to go...
Well just a few thoughts and ideas for you....
basically if your on a budget keep it simple