I want to kill my backyard, advice

You can remove puncturevine (goatheads) with sod cutting, but you want to do it soon because it has a long taproot and it's already growing this year (right? are you already seeing the plant growing? flowers yet?) Once it really gets going, even sod cutting could leave some plants behind.

Using a pre-emergent herbicide will help in future seasons before serious weeds like that get going - if you replace your sod now, you could put down pre-emergent in a few weeks or a month.

Are you going to do this yourself?

If the goatheads have gone to seed, try dragging a burlap sack or some other material all over the yard to snag up the dreaded goathead. Toss it in the trash....repeat as necessary. Those things that stick are its seed, so........

To eliminate the plant--one good way is to fertilize the heck out of the grass you do have and it will crowd out the weeds (and the goatheads). You will have to mow and water more often, but you get a pretty lawn and get rid of the bad plants.

To temporarily kill weeds (and grass) get some white vinegar and throw it on whatever you want to kill.  It's a nice non-toxic treatment for those with pets.  It wouldn't be practical for a whole backyard, but it's a helpful tip for other applications.  The weeds will come back in a few weeks.

I'm feeling the same way about our yard. The gardens are great, but the lawn is a sea of invasive weeds I can't seem to get rid of. By far the worst is lesser celandine that is taking over the yard. I've been digging it out for years, but it just keeps growing back. I may have to resort to herbicide, but I hate to kill off everything else and don't like to use toxics in an area the dog and cats hang out. Any advice?

If you have a small enough area that you can just dig out the sod and start over, I'd go for it. Maybe even hire someone to do it for you so you can get it done faster and not lose the entire early growing season when new grass is more likely to take well. Good luck.

I'm planning on using vinegar on them too. I have them in my front garden and coming up between the bricks in the walkway. I'm worried that anything stronger might discolor the brick, so vinegar it is. I know a lot of seeds fell between the bricks last fall. 

Why? Does boiling it make it more effective or something?