So, I tackled this same conundrum recently, with a heaping helping of ever-present social anxiety to boot. Here's my experience:
I had no idea what to do. I have some friends who climb that I work with, but they are very busy and really didn't seem keen on taking me when I dropped hints. I didn't want to push the issue because I always worry about being a burden and ruining people's experience.
So I stumbled across my local Mountaineering club, and they do classes.
There's no 'certification' or anything but they did give us 4 solid days of instruction and experience. Basically we did a short indoor session of what's what, then just practiced different techniques and anchors at two different climbing areas. You basically need to learn five things to start:>What the types of equipment are>How to do 'buddy checks'>How to belay and communicate>How to build anchors safely>How to use basic climbing techniques
So, I was really happy with the class because it absolved me of any guilt of messing up someone's climbing day. But having all the people around was kind of stressful. Especially since I wasn't (still am not) very good at any of it. This was the first outdoor activity that's really challenged me. Like, backpacking is fine, mountain biking I do OK at, shooting I'm OK at...but figuring out climbing kicks my ass.
I actually had this big existential crisis of sorts over the matter. Between the level of socialization involved and the fact I wasn't picking it up as fast as some I almost quit the class. But, I made myself finish it.
It's not my favorite sport; I really only took this to get my feet wet for a mountaineering school next Spring. But I'm going to keep at it because it really is challenging to me.
Lastly, the class introduced me to people to climb with, the most important part, because you need a belayer. This was actually the most challenging part for me since I'm so introverted. But we are already working on plans to climb together.