<human>: Can you provide a brief overview of the text?

Okay, I'm new to reddit, but  this seemed like a good place to get things off my chest and get some advice.  This might be a bit confusing, but I'll try to be as clear as I can.

I've been friends with two women, Alice and Beth, for around a year.  I've known Claire for almost as long.  Alice and Beth are friends. Neither knows Claire, but we all go to the same university.

Three months ago until about two weeks ago, Alice and I have been together.  Unfortunately, we weren't that compatable and split up, but we're still good friends.  

A few days ago, Beth invited me to a dinner date (hasn't happened yet).  Alice is okay with this.  

Yesterday, I was with Claire, and she told me she had feelings for me, and wanted to become more serious.

I've only had one serious relationship before college, and I don't know what to do.  I really like Beth and Claire, but I think i feel more strongly about Claire. I don't know.
 However, I do value my friendship with Alice and Beth, and I don't want to ruin that relationship.

I was thinking I could go out  with both of them at first and see how it turns out, but it feels sleazy doing that. 
Plus, how can I let one of them down easily, or explain what's going on with the other woman?

 I guess I just want some advice on how to handle this.  I'll try to clarify if I did some formatting wrong or wasnt understandable.
<bot>: do I risk a relationship with two friends by dating one another, or try to date two people at once (Who may see each other with me)?