Well, it's not that simple!
Your kids are probably already canadian because their father is. So for them, you need to apply for their passport, and they should be good to go.
For you, you need to be sponsored by your husband to become a permanent resident of Canada, and be legally allowed to stay in Canada as long as you want. it takes a while to fill out all the forms, and get everything ready, as well as wait for the paperwork to be completed.
You can apply either inland or outland. For US citizen, it's usually best to apply outland, because you can keep on travelling between the US and Canada, and you can start the application process now.
So - read the CIC website about sponsorship, print out the forms to understand what you need to do, and go ahead....
Outland currently is 43 days for stage 1 and 11 months for stage 2 - but a lot of people are done faster than that. Also - because you are from a visa-exempt country, you can go and stay in Canada even if your PR is not issued yet.
So technically - June 2013 is probably too early to have your paperwork completed ( depends how fast you fill out everyhting etc, but seems too short), but you could start now, and at the same time start the paperwork to get the citizenship certificate for your kids, and get them their canadian passport. In june 2013, you can move there, but you will have to go as a tourist if your PR is not issued yet - which means that you're not allowed to work, and depending on the province, no healthcare, until you have your PR. but you could still go.
read the first post by Leon called "spousal sponsorship" it will give you an overview of the process and what to prepare.
then go to CIC website and read all the guideline, and get ready! here is the link http://www.cic.gc.ca/english/immigrate/sponsor/index.asp
Good luck,
Sweden