Is this the beginnings of a relationship?

Let me preface this by saying that I've never been in a real committed relationship before. I've only ever been on a few dates. (I know, I'm super late in the game at age 25)

I've had a crush on this girl from work for a while. I didn't confirm she was gay until a few weeks ago, although I had my suspicions. I met her a few months ago when I started doing over-hire for the company she works for. The day I found out she was gay a bunch of us went out for drinks, and she gave me her number so we could figure out plans. This was the first time we'd seen each other outside of work. Since that day, we haven't gone a day without texting each other after work. We don't act any differently at work. For me it's because the people at work gossip a lot, and I work with my best friend who I'm still not out to. But this girl is the only person I've been able to have a more than 10 or 15 text long conversation with. (I'm terrible keeping a conversation through text) Do normal people do this with friends they just met a few months ago? Is this the beginnings of a relationship? How do I know? How do I proceed? Am I supposed to like ask her out or something? I think she likes me like that, but I can never be sure! Help!!

On a side note, I'm probably going to use her as a segue to come out to my best friend! I hope that goes well. :/

 

Update: I think she just asked me out on a date. We're going to get Ethiopian food next weekend. I don't know how to start a relationship. This is all very confusing. And my depression-brain just keeps telling me that she doesn't like me like that.

 

She's out at work but I'm not. I'm only out to one person who now lives really far away from me.