I've had several girls try to establish a connection with me in some way, either by looking at me a lot, or by coming up to me and saying hi, or by just being very friendly when I start talking to them (responding well to me talking, asking if I have facebook etc). Most of them are very attractive and one's all right.

But then these girls start giving me mixed signals and some of them will show complete lack of interest.

I even had an attractive older woman (who works at a doctors that I have to take one of my relatives to) give me some very obvious signals (winking at me whole saying to call her to tell her about something my relative was talking about, another day winking while saying I could have come alone to drop something off). One time I didn't catch on and the other time I didn't know what to do in the situation. The last time I was there, she sat down and started talking to me as my relative was getting checked up. As we were leaving, I asked her for her cell number (joking mentioning the first signal that she had given me), but she ended up giving me the office's card instead, saying to call her of I have any questions. WTF?!

Is it possible that these are just the type of girls that like to mess with guys, or is it that I'm failing to create and/or establish a good connection with them?
Maybe I just need a lot of work on my communication skills and this is just a game that I need to keep playing, but any advice/input from you guys will definitely be appreciated.

Girls coming up to you have made a choice, they are interested if your social skills are as good as your looks. They expect you to be something extra, if you cannot recover from the shock quickly enough then they will lose interest. You need to be able to act alpha and you need to not seem uncomfortable in any situation! If they flirt with you you have to look like you expect them to. Wink back and then do your own stuff, no biggie.