You will never get rid of ALL the flies in a rural area, but you can reduce them. It all depends on your priorities.
Check your screens, doggy door, etc..
Use "all of the above:"
Set stinky fly traps away from traffic areas to draw them away from where you spend the most time, and away from possible entry points. Set sticky traps closer to areas like doorways, etc., that can be entry points.
I don't have any suggestions for poison; I don't use it, because I don't want to poison my good bugs and other living things. If it fits your scenario, though, look into some of the automatic barn systems; this is one of many:
http://www.greenwoodsporthorses.com/Fly%20Spray%20Systems.htmI know that there are also electronic killers, which I've never used, so I can't offer an opinion.
Move the cat litter to the garage or some area outside the house, and place a hanging sticky trap above it (out of kitty's reach, of course!)
Scoop twice a day.
Do a dog-poop scoop for the yard around the house at least twice a day.
Use fly predators. When I didn't have pastures, I used them in my manure pile to great effect. With the horses spread over really large areas, you never get all the poop up efficiently enough for predators to be that effective.
Get chickens or guinea hens, and let them free range. They love nothing better than to scratch through any areas flies would be laying eggs.
Use flywipes on your dog/cat if they seem to be attracting flies. I've never actually done this, but it works for my horses, and thus the barn.
Good luck!