Just wondering if I can get help please on some suggestions

There’s a couple of girls I like and have made good eye contact with and smiled but that’s it
Should I just say hi, how are you? Then ask if they want to swap numbers?

More recently I got speaking to a girl at a bookmakers, on the horses and I asked who she had put on then she asked me but that was it . how can I have progressed that kind of thing?

I’m quite a quiet shy guy and my type is the shy/quiet type

And I am not sure on really what things to say in general after making eye contact and smiling.
How can I progress it e.g starting a conversation then like making it flow?

I know it depends on circumstances but I don’t feel I’m getting anywhere

Thanks, any help would be appreciated.