Better meat: In my books, that's meat from healthy animals raised in low-stress situations with food natural to that species, no growth hormones, very little use of antibiotics (if animal gets sick it can be treated, but not slaughtered within 60 days of such treatment) or other drugs. Thus pasture-raised cattle, hogs with some room to roam and wallow, sheep on grass. Ideally, the pastured animals are on pasture fertilized only as needed with organic fertilizers. Animals are handled quietly and gently; transported for as short a time as possible in as low-stress a way as possible, and slaughtered humanely. Processing plant adheres to modern standards of cleanliness and safety to maximize the quality of the meat from the healthy animals.
If you can't afford a farm/ranch of your own (and most of us can't), you might make a deal with a farmer or rancher to pasture your stock on their land (for a share of the crop or for money) this is an excellent way to go. I started by purchasing two cows from a local rancher, and trading half the calf crop each year for pasture use. To do this, you have to know something about the animals--enough to pick good ones--or you have to really trust the farmer/rancher. I chose to purchase their cows because they had a certified TB-free and Brucellosis-free herd that had been on the same land for over 20 years...healthy, sound, fertile cattle.
If you go this route, then you are responsible for finding a meat processing plant to slaughter and pack the calf/hog/lamb and for transportation to the plant. I pay the rancher for the gas it takes to drive our calf to slaughter. Still, you will be getting quality meat for less than supermarket price (you just need the storage capacity for it. But you can share it with friends, and if you have a group who agree to pay a share of the cost for a share of the meat, you're all ahead.)
Another route: contract with a producer to buy one calf at market value (whatever that happens to be) and pay for its transport and processing. This will add to the cost, but will probably still be below retail. If I had paid market value for a dressed carcass (estimated at $1.40/pound the week before Christmas), this would have raised my cost from $0.90/pound (processing cost plus gas to transport the calf and gas to transport the meat) to $2.30/pound...which is less than the $4++/pound average retail price of beef.
You can also buy a calf at a cattle auction, but there you know nothing about the calf's history prior to the auction--was it on pasture, or was it being supplemented with grain, or on hormones, or what? I think this is the riskiest approach--if you know enough about cattle to pick a good calf at auction, you know enough to work with a farmer/rancher who will grass-feed your calf for you.
Another source for meat animals is students in ag programs at schools (FFA, 4-H.) Show animals are always sold off at the end of the show season. If they're sold at the show, they usually bring a high price (but not always) but nothing says you can't contract with a family who's got a few lambs or pigs or calves to buy the one that doesn't do well at the show so it's not going to the next show. You want it "relaxed" from the heavy exercise regiment that the animals are put through for showing (to build the muscle that is supposed to make them win), but that takes only a few weeks...you can specify that the animal be fed only natural feed, no additives, or add some grain, but no additives, and ensure that it has had nothing in the 45-60 days prior to slaughter that you don't want. I'm buying a lamb that way...it placed last at the last two shows. It's a Southdown, which means a meaty, stocky lamb, and it's getting past lamb-size...but that doesn't bother me as I like mutton, too.
These approaches benefit small farmers/ranchers and allow you to source your meat closer to where you live, with more knowledge of how the animal was handled in life and more control of what you're eating...all things I think are valuable. Connecting with others interested in local food production and consumption may make it possible for even single people living in apartments in cities to find high quality meat at reasonable prices.