What is a cheap puncture-proof tire against tackweed?
Forget the slime, thornproof tubes, and tire liners.  IMO, the best puncture protection for under $40 are Vittoria Randonneurs.  I've had excellent luck with them.  I had a pair of Gatorskins before the Randonneurs, and trust me, Kevlar belts do not work as well.  Because the Kevlar is a woven material, very sharp objects can penetrate between the Kevlar fibers.  Happened to me at least six times with the Gatorskins over a two year period.  Blackberry thorns, glass, and even staples penetrated the Kevlar.

Also, since you say you're not very versed in bikes, I have to wonder if you may be pinching the tubes when mounting them, or leaving whatever punctured the tube in the tire, allowing the next tube to be punctured as well.  Are you checking for any sharp objects poking through the inside of the tire, or for any burrs along the inside of the rim?  I always try to find the puncture in the tube before removing it from the rim, so I can concentrate on that area of the tire for a closer inspection.