You may visit, but unless you and your fiance can prove one year's uninterrupted cohabitation (except for short breaks for business etc) or actually get married, he cannot sponsor you just yet.
You may visit for up to 6 months as a visa-exempt American, however if you bring your children with you, plus a lot of baggage and possessions, it is unlikely you will be permitted that long, and you might even be turned around at the border. When you go to visit, it must be obvious that you are going to do just that: visit. You cannot move over with your kids and possessions until you are granted permanent residence.
Your children will be required to get student permits and pay international fees, in most cases, to attend school. After they receive PR, then they can attend school without paying additional fees.
You will need actual passports, not passport cards for the PR process, so you should apply for them. If you have sole custody of your children as per court orders, you will need to produce the court order for the purposes of the application. If you do not have sole custody, your children's father will need to consent to their emigrating to Canada.
If you haven't already done so, read Leon's first post in the first thread at the top. It will give you an overview of the process and what you should prepare yourselves for, you and your spouse.