I understand your frugality... but how BIG a yard are you looking to attack, and how BADLY overgrown is the yard? You may be STUCK hiring, renting, or buying a new yard-toy.
My brother lives on a 25 acre ranch in the California Sierra foothills, and we suffer from this problem ALL THE TIME !! And up there at this time of year it wouldn't be just an aesthetic issue, but a FIRE hazard.
During the winter / spring months we actually use all three of the recommended methods:
1) We have two gas-powered weed-whackers for the fence lines and the edge of the road... we use this to do trimming in the summer, because the nylon line won't start a fire if it strikes a rock.
2) We set the riding mower to it's highest setting (about 4 inches) for the first pass on Saturday morning, and then re mow at 2 inches Saturday evening. AND if the grass is VERY thick, we go VERY slow... slower than walking speed.
3) This last winter, he finally bought 6 goats... WOW those little critters do WONDERS... they nibble right on down to the stubble !! They'll also clear out undergrowth... and here in California, they're pretty simple to RENT.
A weed-whacker is a reasonable purchase, and well worth it over the years