Let the mother do all of the disciplining. If you have an issue with a behavior or action, discuss it with the mother when the kids are not around. Don't try to be a father or a father figure to the kids. They probably already have a father and will only resent you if you try to take his place. Offer to pay the babysitter.
Many single mothers struggle to pay the bills and support their families.
Your date may want to see you more often but can only afford to pay a babysitter once a week. Babysitting fees may put a strain on her budget. It would be a kind gesture to offer to pay the sitter for her. Include the kids. After you have been dating for a while and you feel there will be a long term relationship, you should plan to include the kids in some of your outings.