Is it more economical to true a cheap wheel or to buy a better wheelset?
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.