What do you do when you see a new person at church?
I've experienced the range from being completely ignored, to being courted by far too many people.
Here's best, in my view:
If the person is sitting awaiting the service to start politely excuse yourself, introduce yourself, mention that you've not seen them before, but you hope they find the service to their liking. Then move to your seat.
Do not assume they are a Christian, do not assume that they want a long converstion, do not assume that they want a bosom buddy (unless their demenour suggests otherwise).
After the service briefly greet them again, and again, if it seems appropriate, have a brief conversation: perhaps something like:
"That was a typical serivce here, how did you find it?" Without quizzing them (you know, normal polite conduct on your part).
If they want to chat they may open up the conversation, perhaps give them the opportunity by inviting to get them a cup of tea, etc. Offer to introduce them to others. The conversation may or may not go anywhere.
To remember: do not assume they want a long talk. I remember at a stage in my life I would not want a long conversation at all, just a nod and smile of acknowledgement was enough. Others may be similar. Demeanour will tell all.
The biggest thing however, is to avoid the 'evangelical conversational butt' that is the 'butt in' that evangelicals seem to specialise in. If you do need to talk to the other person (the church regular who's talking to the visitor) excuse yourself and ask if you could speak to them afterwards.
I know this will shock many evangelicals who ride roughshod over normal manners and simply ignore the other person so they can talk to the one they know.