Is it more economical to true a cheap wheel or to buy a better wheelset? Truing wheels around here costs about 20 dollars. 

I have a cheap rear wheel that has 32 spokes. This is a rear wheel that came with a Bikes Direct bike, to give you an idea of its quality. The hubs are unbranded and the rims are Alex DA14s. The wheel is severely out of true and I've resorted to taking off the rear brake pads. The front wheel is better off but not by much. 

I could buy a new Shimano wheelset for about 100 dollars. From what I read online these wheels come true and stay relatively true for a few thousand miles easily. My experience with lower-end, big-brand wheels has been wholly positive too - they've all stayed very true and none of them have required re-truing.

I could also spend about 90 dollars and get the Performance Titan wheelset. The thing about these is that they crack after a few thousand miles apparently. But I don't mind since Performance has a lifetime satisfaction warranty, so even if they fail, I can just exchange them for a new one, free of charge. 

Would it be worth buying the new wheelset and running them down into the ground or truing up my cheap wheels? To me, truing my cheap wheelset once gets me halfway there to a new wheelset that likely won't require any truing during its lifetime. What do you think?
Whatever you decide based on answers here I suggest you get a decent, dedicated spoke tool and print off or even buy on paper a comprehensive guide to truing wheels. You may have a multi tool with a spoke slot. Pretend you haven't. Use the brakes pads as a guide and hang the bike off a car rack or put it upside down, rather than spending a lot of money on tools which you might not use again. A spoke tool is cheap and handy for replacing the odd broken spoke. 

If all goes well, you've got true (or true enough) wheels and you've learnt a skill. If you get the wheels true, and they won't stay true even though you've relieved the twist in the spokes you've got your answer and you can buy new wheels. If you find you can't improve matters enough, you're back to choosing new wheels or taking them to the shop. But you'll be better placed to make that judgment once you've developed a bit of a feel for it. It sounds like they're bad enough for now that you can't make matters significantly worse.