The most important thing before dating is to be sure you are really DONE with your ex - that you wouldn't take them back even if they came crawling back. It's just not fair to date people and then end up reconciling. But it sounds like you are safely there.
As for the schoolteacher - the thing I'd be most concerned about it, does she want children? And if you did date a woman who wanted to have kids, would you be open to it? If not - if you feel yours are enough and you don't want to father more kids - you should avoid dating childless women unless they make it clear they have never wanted to have kids.
And please, don't introduce your kids to your dates until you've been dating at least 6 months and are pretty sure it is going somewhere. Kids don't need to go through the process of getting emotionally attached and then experiencing another loss when the relationship breaks up.
