Gathered that, sorta got the hint on the 8th rejection/ apology
Apologies for my previous message it was harsh and of no use to anyone.
Now that I've had a rest and have lost the headache, I'll try and give a better answer.
The main thing that came across from your post is that you've got low self confidence. This feeds through everything you say, and will be affecting your actions more than you will realise.
I know this, because for a long time through all my teens and early twenties (still am to a large degree) I was the same.
From the situation with the girl, you've proved that you're not a weirdo, that you can hold a conversation and interest so take heart from that. The fact that she wants to go on a trip with you is also positive.
However, in this particular case, I would not say that she is after you romantically and if anything was to happen, it would be brief and awkward!
I would gently say to her, "thanks but no thanks", and explain that you need time to cool off from the whole thing, cite the conversation before.
It seems like she is in need of someone to vent to, someone she does not perceive to be a danger and who she can release to, before going about her way to the next fellow. Everyone likes to feel liked and wanted (she probably got **** upon by the last fellow), and she know's she'll get it in a safe way from you.
Use the experience of this to realise that there are MANY girls in the world (seriously there are loads!), and that knowing that you're well within the statistical distribution curve of blokes, means you're down to the same game as all: the numbers game. For girls worth having, you don't or shouldn't need to badger or "win over through repetition", it's a challenge but not a concession on anyones part.
Get started with your self improvement, build yourself up for your own confidence, start spending more time reading and doing different things, evaluate what you are all about, and you'll find that over time the confidence will come to you, and as a result you will have a far greater degree of success, but ultimately more importantly you'll be happy, and will be more likely to make someone else happy.
I'm not sure on your age, but it's taken me to my mid twenties to realise this.
Good luck!