My sense is that it's not getting gas - I try to give it some throttle and nothing happens. ...
...my brand new battery dies even after riding it around. ...
...until last night it fired up on the first or second kick. ...
Clarity please, on conflicting statements:
"...even after riding it around. ..." implies it has been successfully ridden. Can we assume it ran well at the time? If so, retracing steps between then & failure to run could help.
We personally don't know KZ400's well at all, but it hails from the same era as our '73 through '75 Z1's. They have three positions on the ignition switch; OFF, RUN and PK (parking). The PK position illuminates only the tail light, and the key can be removed in the Pk position. It's easy to miss the tail light being on if one doesn't walk behind the bike & look, and the tail light will drain the battery over night. We've done it. Perhaps the ignition was inadvertantly left in Pk position?
Troubleshooting bikes can be challenging using text based means such as forums. Appreciate that you're willing to learn not only to ride, but also to learn your bike hands-on. We can help you get through this. Just resist becoming discouraged as best you can.
The Lovely Z1BEBE sez 'Hi! Nice to see another woman here!'.
Good Ridin'
slmjim & Z1BEBE