We have quite a few goatheads here in SE WA. Just yesterday while tooling down a sidewalk next to a vacant lot I suddenly realized I was in a minefield -- picked out a half dozen goatheads from my tires. No flat though

Thorn tubes are the single most effective thing you can do. Not 100% effective, but only every 10th-20th one will get through instead of every one. Slime or the equivalent will fill most of the holes. Yeah, it's heavy but not as heavy as pushing your bike home.

Kevlar tires with the thorn tubes are about the best you can do. That, combined with watching where you go is about as goathead-proof as you can get.