If I dated a single person with no children, hed have no responsibilities towards my child unless we got married. Then Id expect him to care about her, and to be a good role model. The same way if I met a single person with child/ren. Id get to know them if we were getting serious but I wouldnt be obligated to support them financially etc until theres a ring on the finger.
But I think most men and women who care enough about the person they are dating, are willing to get to know the kids early on because they are part and parcel of the single parent.
Now that I think of it, my former sister in law had kids to my brother. They got divorced and she met a new guy. He told her to let the kids go to my brother so she can be with him. They ended up getting married but she was so torn between her kids and the new husband that it not only destroyed her 2nd marriage but also damaged her kids.