I have a back yard I need to tame. Lots of Kudzu and small brush (< 1/2" diameter). Growing up we always used the old fashioned string trimmers. However going to shop now I see very, very few "string trimmers" and lots of "power heads" which you have to buy attachments for. Which is better to purchase for clearing?
(Poison is not an option, I'm strictly concerned with manual removal).
You have several options.
Gas trimmer (no power head option). These can be a bit cheaper than a power head option, but have less versatility, you can't convert them to a brush cutter, limb trimmer or rototiller. They might also have more durability as they don't have to worry about the connection between the power head and the attachment. There aren't a lot of these available anymore, most of them are power heads with string trimmers attached by default.
Gas Power head with string trimmer. If you need a trimmer anyways this might be the way to go here. You can start with heavy duty line, and if you need more power you can pick up the optional brush cutter attachment that gives you even more cutting ability. I went this route and recently bought the rototiller attachment and while it's not like having a real rototiller, it's reasonably effective for the investment.
Something heavier like the DR Trimmer - If you're going to have to go through this dance every year, you might want to invest in a push string trimmer with optional blade attachments. DR Trimmers makes one that looks really nice.
Personally unless you're expecting to do this kind of massive cleanup every single year, I'd go with the string trimmer and get the brush cutter attachment, it's going to be cheaper (you can do that for ~200$ if you go curved shaft on the trimmer) and it will likely meet you needs, although it will be a lot of work upfront. However, if you're not planning on maintaining your backyard and letting it overgrow again than putting down the cash and getting a walk behind string trimmer is the way to go.
Getting rid of kudzu can be a challenge...but the best and clearest way is to kill it's root crown. It's difficult, BUT FAR less time consuming than herbicides and constant pruning.
Find a vine lead that leads to the ground. Start digging around it without cutting the vine. ONCE you've dug about a foot, you'll notice a large Woody piece, that's the root crown. 
destroy it, cut EVERY woody root about a foot away from the crown in all directions. Once the root crown is harvested, destroy it by burning. While this takes a lot of hard work and muscle, it is far less time consuming than pruning, herbicides or chemicals.
With the crown destroyed, the vines will die and brown off, unfortunately that makes them a fire hazard, so start trimming the kudzu vines instantly away from any property
