Congratulations on your husband's job offer.
You need not worry about your different nationality. You as a Polish citizen will be equally welcome in NZ as your UK citizen family-members.
You need to look at the Immigration NZ website, and follow the instructions for whichever kind of visa your husband will be applying for. Here's a page with ways in to the different kinds.
https://www.immigration.govt.nz/migrant/ It's quite likely that the employer will want him to apply for a temporary work visa to start with (because these can be processed more quickly than Residence).
http://www.immigration.govt.nz/migra.../requirements/ If he's applying for one of those, he can sponsor you to apply for a partner-sponsored temporary work visa at the same time,
http://www.immigration.govt.nz/migra...milystream.htm, and student visas
http://www.immigration.govt.nz/migra...yinnewzealand/ for any of your children who are five or over, or visitor's visas for those under five, at the same time. (You send everything together with a covering letter, and the official will work it all as one case, supplying info like your husband's visa number when it becomes available.)
When your husband applies for Residence, he will be the principal applicant, but for that, you can all be secondary applicants on the same form.
http://www.immigration.govt.nz/migra...nt/default.htm
As to the police checks, you're going to need one from your country of origin, and one from any country you have lived in for more than a year during the last ten years (so, the UK, definitely, from what you've said). See the details here
http://www.immigration.govt.nz/migra...cecertificate/ about how to obtain the certificates from the two countries.
Don't worry. All the information you're going to need is available on the INZ website, and if there's anything you don't understand, there is usually somebody here who can help explain. The INZ website looks intimidating at first, but if you take your time and make sure you're looking under the right heading, it's possible to get your head round things quite quickly.