Enough of the "Eager to Please"
When you sense that the girl is becoming friendlier to you, it is time to change your approach. In a FIRST PHASE approach you may have had to grovel a bit to get to meet her. You said, "Excuse me." and "I hope you don't mind" etc. This may have been necessary. After all, you were intruding in her space. But now it is different: she has accepted your presence.
Now you should act in a more self-assured fashion. You should no longer act as if you were only eager to please her. You should now act as a man who is confident that the girl is (or will be) attracted to him, because he has something worthwhile to offer. And you now expect to deal with her on an equal basis. Your attitude should no longer be so apologetic, but more as if you were taking her approval for granted. Your speech style should reflect this:
Do not say, “Would you come with me? I'd like to show you something." Say, "Come with me. I'd like to show you something."
Do not say, "I hope you don't mind my asking, but what do you think..." Say, "Tell me! What do you think?"
It is better to sound a bit selfish and demanding than apologetic and unsure.