There are many ways to handle it. The most important thing is to establish how you and your new spouse will handle it BEFORE you remarry. It is not a good idea to remarry until you resolve all concerns and the two of you agree on policies regarding the children as part of one family, not two.
You cannot expect his kids to respect you as a mother and you can't expect yours to treat him like a father. Don't try to parent each others kids. It's going to take time for them to adjust. For now just parent the kids that are biologically yours.
The best way to blend 2 families with children is for both parents to treat all children the same. All children must feel as if they are equal in the family. Neither parent can have favorites or treat their own children any different from the other parents children.
discuss expectations with extended family and be ready if/when they are not fully supportive to your "equality" requirement towards kids. your kids shouldn't lose over your selfish desire to be together. figure out how to deal with it without harming the kids relationship with loved ones or don't marry.