We moved quite a bit when I was young, so I went to a lot of different schools. To make things worse, I was painfully shy. If you've ever been new in a ward, you know a little about that awkward, unsure feeling. Even if you've been a long-time member, there's always the unknown when you're new. Maybe you're not sure where the Gospel Doctrine class meets, or what the Relief Society President's name is, or what day the Cub Scouts meet... Its unpleasant at first, but you usually pick up the details of your new ward in a short time.
Imagine, if you will, how much more awkward being "the new kid" would be as a non-member visitor, returning less-active member, or new member in your ward. If you're lucky, you're in the kind of ward where people are falling over themselves to meet newcomers and want to become lifelong friends with everyone. Unfortunately, that just isn't happening in most wards. I really don't think its ever intentional. Some people are reserved/shy. Many are so preoccupied that they don't even realize that there is a new person at church. Worst of all, many members assume that "someone else" is taking care of it. (Oh, that unreliable "someone else"...they never seem to follow through!)
I've been in a lot of wards. I've been in wards where nobody seemed to care that our family was there. It probably wasn't true, everyone probably assumed that "someone else" was greeting us, showing us around, and befriending us. One particularly large ward we briefly attended had just been created from combining two smaller wards, so most people didn't even realize that we were new to the area. Half of the ward were strangers to the other half anyhow. Everyone seemed to have friends already (from their prior ward) and for us it was an extremely lonely 9 months (and we are active members!). I remember that I burst into tears when our home teachers called us for the first time, asking for appointment to visit us. They called the week we were moving away. We could have really used some friends during the previous 9 months, but by then it was too little, too late.
So let's all resolve right now to never let that happen to another one of our brothers or sisters. Everyone should feel welcome, ESPECIALLY newcomers, who probably feel especially conspicuous and vulnerable. If you see someone you don't know, they're probably new. There is always a possibility that they aren't new, which could potentially be embarrassing. And we want to avoid embarrassing.
Imagine this scenario:
"Hi, are you new?"
"NO. I've been in the ward for 5 years. I think you're my home teacher"
Well, I try to avoid awkwardness and embarrassment in general, so here is my approach to this situation. When I see someone I don't know (sitting near me in class or in the chapel, standing out in the lobby before church, sitting alone at an activity), I say something like, "Hi, I don't think we've met. My name is....." Thats step 1. After that, without fail, they always introduce themselves to me. Usually they just tell me their name and don't really offer any additional information. So I move on to step 2. I ask, "Are you a member of the ward?" which I quickly follow with "I'm fairly new in the ward, so I don't know everyone yet". This keeps things from getting awkward in pretty much every situation. If the person is new, they may be relieved to hear that you don't know everyone either. If the person is a long-time member of the ward, they won't feel embarrassed that you don't know them because you're "fairly new". If the person is less-active, they usually are quick to say so.
I used this approach a few months ago when I saw a young mother (who I'd never seen before) trying to keep her child occupied in the hallway during Sacrament meeting. When I had introduced myself and mentioned that I was new, she said, "Well, we haven't been very active lately". Its hard to know what to say to that, so I tried to change the subject slightly. In my most cheerful voice, I said something like, "I haven't been to church very much lately either because I was on bed-rest with my pregnancy. How old is your baby?".
Now, I AM "fairly new" (less than 1 year) in my current ward, so that "excuse" works fine for me right now. There are probably dozens of other reasons you could use. "I have a calling in Young Womens, so I don't know all of the adults", "We've been doing a lot of traveling, so there are a lot of new people I don't know", "I'm usually away visiting other wards for Stake Primary, so I don't get a chance to meet many people in our ward", you get the idea!
If it helps you, you can also play out several of these "Have we met?" scenarios in your head to get practice. You don't need to memorize anything, just get the general idea.
So there you have it, a painless approach to introducing yourself to a stranger in any church-related setting. I don't mean these "lines" and "excuses" as trickery at all, so please don't take it that way. I've read several books about communication and presenting yourself in a positive way. To me, these are methods aren't meant to deceive anyone, but rather to help me to be more outgoing without worrying so much about being embarrassed by saying the "wrong" thing. Its a stepping stone to help you start a conversation.
Even the simplest conversation can make the difference between a good first day in a new ward and a bad one. Don't wait for "someone else" to do it!