Sound System on $10K Budget

A subwoofer (or two) paired with stand mount speakers is nicely flexible.

I have recently built a system from scratch and have paired a set of Fritz Carbon 7 SE Mk2 speakers with a single F12SE Rythmik Sub. 

Rythmik Subwoofers: https://www.rythmikaudio.com/

Fritz Speakers: http://fritzspeakers.com/

The powered sub (s) relieves a lot of burden from the power amp.

I went with a Van Alstine SET 120 Control Amp to start with, but then added a Van Alstine Transcendence 10 Tube Preamp when I got the sub. I run the signal from the preamp into the sub, then using the "High Pass Filter" from the sub, I go back to the power amp, which drives the speakers.

AVA: https://avahifi.com/

I'm very happy with this system and you might enjoy reviewing those brands.

As stated by others, 1st find a pair of floor standing speakers that can play into your large space at the volume level you desire. Stay away from those having low sensitivity and impedance. 

Look into the purchase of a 2 channel integrated amplifier that has subwoofer output and a power rating that will easily drive the speaker of choice.

You will need a DAC to take digital signal from TV and iphone. I prefer a stand alone one. That said, many integrated amplifiers have built in DACs. Make sure DAC has the required inputs. 

Blue Jean cables make for a good start at modest price. 

I believe 10K is more than enough budget to get to where you want to go. Buying almost new but used saves. 

Thank you everyone for all the help here! Much appreciated. There are so many factors to consider when putting a system together, so every post has been enlightening. Putting together a headphone setup is a walk in the park compared to this. 
I will keep looking up loudspeakers as a starting point, and will come back with updates.

1. I think 10K is the perfect budget, easy to find components that pair well

2. the 5K budget for speakers has the widest selection - in the standmount/bookshelf category

3. I would buy new speakers, used amp and a new source

4. I would stay away from the Ferraris of any component such as SF speakers and Mark Levinson amps, you pay a lot for the brand/image when you go high end

5. Don't buy the cheapest cables, a few 100 bucks will serve you well

6. When you audition systems, start with the cheapest combination and check out better speakers gradually

Good size room. Bookshelfs will work not just floor stand speakers.  Stay with my rule:

40% = speakers

30% = amp

20% = source

10% = preamp

Cables mean nothing.  Those people paying 1k on wires and stuff is silly.

My favorite system is the NAD2200 wth 1155 using Elac Unifi UB5.1, total.cost me $600 bucks.

The sound difference between 1k and 5k is barely noticeable. To 10k. Some differences, but small. I have vintage and new stuff to compare.