Round-up isn't going to do anything for very long. Goat's heads are natures way of having the seeds spread a long, long way. The plants spread out like a mat and cover large areas.
for dogs, you just have to learn to spot the plant. Our dog always stops immediately and pulls up her paw, and we pull it out. The trick is to not keep waking thru the stuff. Out here it grows from landscaping strips onto the sidewalk a bit. When walking in sandy areas, just avoid the plants, and you'll avoid the majority of the goat's heads.
If you've run into a lot of them it's real important to check your shoes before going into a carpeted area. Stepping on one with bare human feet is a real experience!
For bikes, you need two things - a protective tire liner called "Mr Tuffy's" AND thorn resistant tubes. Either one alone won't cut it. If you do get a puncture, after patching the tube you have to take the tire off and Mr. Tuffy's out, and thoroughly run your hands inside to catch the points that broke off on the OUTSIDE of the tire, leaving a little sticker on the inside to puncture your tube again.