It is amazing: everywhere the water spilled, pigweed is growing now. Pigweed is quite distinctive because of its pink/purple underside.
Its purple hue gives it away: just like its magnificent large and purple "tame" cousin, grain Amaranth, pigweed is in the amaranthus family. The two common pigweeds are "careless pigweed" and "tumble pigweed". My advice: get the pigweed while it is young.
As it gets bigger, pigweed stems get really tough, making them hard to tear down or tear out later. This is especially true of the "tumble pigweed".
Not to be confused with the Russian Thistle, which is the true "tumbleweed".
I keep my main gardening area bare ground, so I can just scrape a hoe over the topsoil, killing them while they are seedlings. I just leave them there to decompose on the ground, which has resulted in a nice layer of topsoil as well.
The common purslane remains soft as it grows, so it is less of a nuisance later. Sometimes I will even let it grow quite big for awhile, letting nature provide me with some good compost biomass.
The "prickly lettuce" weed will be big and seed-headed now, if it survived as a garden volunteer from your spring planting.
Lots of "puncturevines" will pop up in your watered areas now too. As you can tell from the name, you really don't want to let "puncture vine" mature. They produce those "sharpy thorns" that make your kids cry when they step on 'em.
The UA book has this to say about "puncturevine":
"It is abundant, one of the most obnoxious weeds in southern Arizona, and is found throughout the state, principally in July and August. Each plant produces innumerable burs ... Home owners and their dogs probably dislike puncturevine more than any other weed because the stout spines can easily penetrate shoes, bicycle tires, and dogs' feet."
If you have an Palo Verde or Palm trees nearby, there is a good change you are getting tons of their seedlings pop up as well. Rip the Palo Verde seedlings out quick. As soon as they get just a little big they get barbs all over 'em.