Ooooo. I approve of these kinds of gifts
😀
First the easy stuff:
For speaker wire, quality copper is all you need. It should read OFC (oxygen free copper) and you can get it many different places, don't get CCA (copper clad aluminum). Monoprice, Blue Jeans Cables, Amazon, Radio Shack, chances are your local hardware store has it too among other places, you shouldn't need to pay more than 50-70 cents per foot. Just make sure it's of sufficient gauge for the run;
http://www.roger-russell.com/wire/wire.htm#wiretable
Basically take the ohm load your speaker presents, length of run and it gives you the minimum gauge you should use. It's usually cheapest just to get a bulk spool for everything, instead of one spool for the front and another for the surrounds. A lot of people just get 12 gauge regardless, I myself tend to get 14 gauge since I never have runs more than about 35 feet. Also, if you plan on running it through walls, make sure you get cable rated for it for insurance reasons.
Next, yes you'll want a good sized area rug. You have a lot of very hard surfaces, this will result in a lot of reflections and reverb which you don't want. It doesn't need to be a thick shag rug, just don't get the really super thin ones, be sure it covers a good portion of the floor between you and the speakers.
You'll want to have some things that help with reflections off the walls too. Curtains, upholstered panel decorations or actual acoustic panels. You don't need to cover the entire wall, a couple things up front behind your front speakers and then on the side walls about 1/4 to halfway to the sitting area. This is something you can wait on and play with after you have your system running. You can used some sheets or blankets and see if it's something worth doing and what works best, this is something I strongly urge you to do as unwanted reflections are one of the largest problems people have though they attribute it to their equipment instead.
As a side note, you don't want to get rid of too much reflections either since it'll make the room sound dead. Usually behind the speakers and first reflections are what you want to take care of if you don't plan on going all out. Bass traps are also a good thing to read up on and think about.
Now for the not so easy stuff (but more fun):
AVR, it should meet the features and capability you need. More info on what you want is needed before any recommendations can be made. Things to consider: Connectivity; number and types of input such as HDMI, component, composite, also if you want say bluetooth, ethernet/wi-fi for audio streaming or remote control. 5.1, 7.1, or greater. Power required, which depends on the speakers chosen. 3d or 4k passthrough? Atmos/dts:x? Room correction? Multiple zones? Pre-outs?
Not to worry, it's not as daunting as it seems. Budget will probably decide some of it, and some things are becoming more common so you'll get it regardless. Main thing is inputs that you need, if you need 3d and 4k capabilities and 5.1/7.1. For 7.1, you'll want a little bit of space between the listeners and the rear speakers..at least a foot or two.
http://www.howtogeek.com/137896/how-to-place-your-speakers-to-maximize-your-home-theater-experience/
This is a decent guide for ideal rooms and gives you something to work with when considering if you have the space for 7.1. Usually 25x20 is more than enough room but it appears you aren't working with the entire space due to the stairs(it'll help to know about how far the listening area is too).
Speakers and sub...even harder. Speakers are personal preference as for how it sounds. Given that subs handle just the lower frequencies, they don't have a particular "sound" to them so they'll be based more of performance than preference to it's timbre. Given that this is a gift, we'll probably want to start with 5.1 instead of something like 3.1. A majority of the content out there is 5.1 (if it's more than just stereo) so it's not a bad way to start and you can add rear speakers later if you desire to. I'll work around $1600 give or take for the setup leaving the rest for the AVR and accessories.
Probably the biggest decision is what you'd like out of your low frequencies. As I mentioned before, room size is a factor and the larger the room, the more sub you need to keep it balanced with the mids and highs. You have a large room, around the 2000 cubic feet mark is sort of where you go from looking at the $500-600 and under sub to over that amount and you're sitting around 3200-3500 cubic feet.
Not everyone cares or wants that low end extension. It's that really deep rumble that's usually more felt than heard where you are looking at under 25-30Hz. Some people say it's not right unless the neighbor's foundation is shaking during explosions. Hehe, and obviously there's a more balanced middle ground too.
If you really want that low end extension, you're looking at getting a sub in the $700-$900 range with the idea that you'll get another one further on down the road as a future upgrade. It's not going to shake your neighbors, but it's not going to feel lacking either. And who knows, maybe you'll find out that you're happy with just one capable sub. If you know you don't want that really low end then we can dial it back a bit and go with something that's a bit more efficient above 25Hz.
And this really depends on what's on sale too at the time of purchase. Speaking of which, when do you plan on buying?
I'll have to cut this short as I'm leaving in a few minutes, but your thoughts on this will give a better direction for me to find suggestions.