I design and install home cinemas for a living. This isn't a pitch, you're too far away from me to support effectively unless you're going seriously high-end (£50,000 - £150,000). You should start by making some key decisions.
1) What's the focus of the room? By that I mean is this a dedicated cinema because you're a big film fan, or is this a chill-out room where you/family/friends will watch movies and sports and do a bit of gaming?
2) When you decided to dedicate a space in the house as a cinema was one of your first thoughts "How big a screen can I get?!!!" or "I wonder what the picture quality will be like?"
3) Do you see this as a DIY project or more of a collaboration with people who have specialist skills and knowledge
You'll notice at this stage I haven't asked about budget. That isn't an oversight. The answers to the questions above will give an indication whether you're focussed on quantity or quality. At this stage that's more important than imposing some imagined boundary figure on a project that hasn't yet been properly defined.
Projectors and screens: Regarding equipment, by and large it's true to say that you get what you pay for; so no big surprises there. The last 10 years have seen the entry-level price point for projectors drop to sub £500, but that's not quite the same as saying £500 is all you need to spend on a projector. I'd also point out that as product becomes more accessible then it pulls in a lot of buyers who are easily impressed by a small and inexpensive box throwing a large and colourful image. Those people post some enthusiastic reviews but it's fair to say they probably lack a little objectivity.
In very broad terms entry-level projectors are in the £800-£1,500 category. They do the basics of lighting up a big screen with a decent resolution image and have the scope to be improved significantly with image calibration. Some of these are aimed at the dedicated home cinema market where black level and colour accuracy is more important than outright light power. Others are designed for use in higher ambient light for gaming and TV viewing where raw light power is more important than picture fidelity.
The next price-band up is the start of the dedicated cinema projectors. These are not as powerful but have better video performance (higher contrast, better blacks, possibly better video processing for smoother motion handling). These are in the £1,500 - £4,000 range.
From £4,000-£8,000, and then £8,000-£15,000 you'll find refinements in fidelity, better lenses (sharper picture), the inclusion of panamorphic lenses to display a full resolution 21:9 image, and the start of projectors with native 4K resolution.
At £15K and above you're looking at projectors with optical light paths and lenses so good that even when they're out of focus they're sharper than most sub £8K projectors. It's true to say though that the law of diminishing returns applies.
If you've never seen a home cinema projector demonstrated in a proper viewing room with good surround then the chances are that an £800 will blow your socks off and you won't be able to imagine how the image can be any better. A decent dealer will be able to answer that with an effective dem of a higher quality product.
Projection screens are an integral part of the optical quality chain too. At a basic level you're simply looking for something that lies flat and even, and reflects with reasonable efficiency, and that doesn't cause too much of a colour shift as the light hits it. That sounds easy to accomplish, but the majority of budget screens are little better than roller blinds made from fabric or vinyl. They rob the picture of vibrancy and soften the image. This is less of an issue if all you need is the equivalent of a sports bar in your basement and the projector has a tonne of light output to start with, but if you want greater fidelity then the better quality screens have surfaces designed to avoid those problems.
Time is short here so I'll have to leave audio electronics, speakers, and the room (acoustics, noise isolation, light management), and control systems to later.
EDIT _ addition
Audio electronics and speakers: I'll deal with speakers first as their size and type will determine quite a lot of the character of the sound, and as is often the case, the type and size commonly chosen not by the sound but by the looks.
If we presume that we're dealing with well made high quality speakers as opposed to large cheap boxes, then in very general terms the larger the speaker the broader the range of frequencies it can reproduce and the more realistic it sounds. From that you can conclude that a sub/sat system where the centre, fronts and surrounds are very small cubes isn't going to sound as good as a system where the centre speaker alone weighs 10 Kg and is the size of a small suit case. This is Physics 101.
The other advantage with larger speakers is they're often more efficient. That's very important because it's cheaper to buy speakers that are better at converting electricity in to sound than it is to buy the extra amplifier muscle required to compensate for inefficient speakers. You'll see speaker efficiencies (or sensitivity) quoted in dB or dB/W or dB/W/m. dB is decibels. W is Watts. m is metres. (Tip: if there's no quoted dB figures then ask yourself if the manufacturer is hiding something.) The centre speaker from a good full sized surround system will have a sensitivity somewhere in the region of 90dB/W/m. Most of the industry sets up its sound systems to produce a sound level of 85dB at the main seating position from each speaker. We call this the reference level. If I swap the large centre for a small sub/sat kit centre speaker then that will run at a lower sensitivity. 84-85 dB is pretty common with decent brands. To make reference level at the seating position with a lower efficiency speaker I'll need to drive it harder, and that means using more Watts from the amp. The 5-6dB drop in efficiency means that I need roughly 4x as many Watts just to make the same sound level.
Now compare that 4:1 figure in receiver power and cost. A budget AV receiver might claim to output 100W per channel and cost around £150. A high-end receiver might claim 200W/ch and cost £2000. That's only twice the power yet 13 times the price. Of course the comparison isn't a level playing field. The bigger amp will do a lot more than simply being more powerful. But the point is that making amplifiers far more powerful is an expensive business. Thankfully there are other factors that mean our amps don't require huge power outputs to produce acceptable listening volumes. However, wasting power on inefficient speakers does have an effect on the amount of reserve power left to deal with sudden increases in volume. We call this Headroom. If you'll allow, I'll use an analogy: Driving a heavy car with a small engine leaves very little headroom. The engine runs pretty much flat out just to keep things moving. There's little or no oomph left for overtaking. The two solutions for creating more headroom are either to put in a bigger engine (more Watts power) or to lighten the car (increased efficiency).
