Omg you are gonna love the Z6! It is such a fantastic camera for family stuff. Basically, that blurry background effect you want is called bokeh, and it is caused by having a wide aperture. Just look for the lowest f-number your lens can do... like f/1.8 is gonna give you way more blur than f/4 or f/5.6. Since you need it by next week and want to stay under $600, here is what I would do:

• Get the Nikon NIKKOR Z 50mm f/1.8 S. Seriously, just do it! It is widely considered one of the best 50mm lenses ever made by anyone, ever. It is super sharp and the way it handles light is just gorgeous. It usually goes for about $530 to $600 depending on where you shop and it is super light to carry around.
• If you want even more of that dreamy look, try to find a used Nikon NIKKOR Z 85mm f/1.8 S. It might be tight on the $600 budget but sometimes you can find a steal on used gear sites like MPB or KEH. 85mm is perfect for portraits because it makes faces look really flattering and the background just melts away. One thing to remember with kids... they move fast! Keep your shutter speed high so they dont come out blurry. I usually try to stay above 1/250s when kids are running around. Just remember that with these prime lenses you cant zoom in or out with a button, you gotta walk closer or further away yourself. It takes like five minutes to get used to it tho. Have an amazing trip!!

Had another look at your post and wanted to add a bit of caution. Like someone mentioned, that wide aperture is what you need for the blur, but be careful about autofocus speed with kids. I remember trying a budget lens once to save money and it missed every shot of my son running around... it was a total disaster. I would suggest looking at these two:

• Nikon NIKKOR Z 50mm f/1.8 S: The safe bet for quality. Very sharp and fast, tho it might push your $600 budget if bought new.
• Nikon NIKKOR Z 40mm f/2: Way more budget friendly and easier to pack, but the blur isnt quite as creamy. TL;DR: The 50mm is the reliable choice for perfect shots, while the 40mm is the practical travel pick. Just make sure to test it before you leave!

Unfortunately, the standard kit lens is not as good as expected for portraits because it wont blur the background much. I had issues with mine looking way too flat. To get that blurry look, you need a prime lens with a low f-number.

• Nikon NIKKOR Z 40mm f/2
• Viltrox AF 85mm f/1.8 Z II The 40mm is super easy and fits your budget perfectly!