New Kid on the Block, Feeling Lost

So I just moved to this new city, and honestly, it's not going as smoothly as I hoped. I thought everyone would be super friendly like they are in those movies where the new kid quickly makes a ton of friends. Instead, every time I try to talk to someone at work or the grocery store, my mind goes blank, and I stutter like an idiot. Today, I had planned to attend that book club meeting I found on Facebook, but as soon as I walked in and saw everyone chatting away like old friends, I got so nervous I pretended to have left my phone behind to avoid staying any longer than necessary. Maybe it's just me being overly self-conscious, but I'm starting to think maybe small talk is harder than it looks! How do you guys handle those awkward moments when trying to make new friends? Any tips or stories about your own experiences in a new city?