First of all, and you will hear this many times on these forums...
If at all possible, ditch the ceiling speakers and put speakers where they belong. I would say in-wall speakers are a MUCH better option than in-ceiling, and if this is a purpose-built room, then treat it as such. Stop trying to make every room of your home the family room. Don't you already have one of those? Make your game/theater space a game/theater space from the start and don't compromise on it unless you really don't care about what you will be giving up.
That out of the way...
A ENTRY level system. The absolute cheapest you get, is a few hundred bucks.
To get 5 speakers around the room for a 5.1 audio setup, will be $150 pretty much minimum for weak speakers. Better in-wall/ceilings will be about $250.
A proper subwoofer may start at under $100, but they are not going to really deliver any bass at all in that price range.
Then you will need a 4K capable AV receiver to run the show, and those start at $250 for the cheapest Denon/Yamaha models.
Add a projector $700/min. + a screen. I would consider a minimum budget $1,500-$2,000 for this.
If you are building a home - RIGHT NOW - then there are things which really are worth spending money on right now, and things to consider.
1. Your speaker wiring should be done properly and accurately. Builders SUCK AT THIS! You will need to be the one to direct their actions on this matter and you need to be on top of them to ensure they are doing a good job. This means that they should use nothing short of 14 gauge speaker wire to your 5 main speakers (if that's all you want) and that those speakers are in the proper location.
2. You should decide NOW where your equipment is going to live. Basements offer a lot of unfinished space or other areas which may get sensitive electronics out of the main space, which can be good.
3. If using a ceiling mounted projector, you should decide on a screen size NOW, and then narrow down some projector options. Then you will need power and a pathway to install a 4K capable HDMI cable into. PATHWAY! Do not install a HDMI cable yet. Be aware that HDMI cables did not exist 25 years ago and they probably won't exist 25 years from now. So, pathway to add/remove cabling is far more valuable than just installing a cable. Conduit of at least 1.25" without any sharp bends, is a minimum requirement for projector locations.
4. Projectors don't do well in ambient light. If you know where you want your screen, zone the lights in the basement up so that lights nearest the screen can be powered off separately from other lights in the room.
5. Builders are clueless.
6. Give this SERIOUS thought right now and get with the builders NOW to ensure the final layout actually works. It can (literally) save you hundreds if not thousands of dollars in rewiring work in the future if you have it wired properly right now.
7. Builders are clueless.
8. Not sure what '6 speakers in the ceiling' means. 5 in-ceiling speakers (bad), plus what? Typical surround systems do use 5 main speakers. Left, center, right, surround left, and surround right. Plus a subwoofer is added. Subwoofers are most often square cube like boxes placed near a corner of a room to add the low end. Subwoofers require different wiring (coax cable vs. speaker cable) and a power outlet nearby.
9. Builders are clueless.
There is a LOT more that people put into their theater builds. Like, a TON more. But, during construction, the most important thing you can do is educate yourself as early as possible on what is out there and make damn sure that the money that you are paying someone else for is being used properly. You are paying for that speaker prewire... don't let them screw it up. But, if you don't deal with it before they wire it, then they will charge you to redo it for darn sure.
If you can post the prints of the space, you will get more information from forum members who are happy to assist.
