if you talk to them like 10 times in a row they will say they need alone time. i try to stop talking when they repeat dialogue, or show a different greeting like, "you again?"
Yeah, if you talk to them enough times repeatedly they'll tell you they need to think about something and walk off with the thought bubble emote above their head. Talking to them again for a little while will only get rejected because they're still thinking about something.
Just got to wait it out after that happens.
In WW they got sad or angry instead, which made me laugh when I found out it was just thinking in NL. Everything got softened up.
If you consistently try to talk to a villager for a good amount of times, they'll eventually stop wanting to talk for a while. Some will appear in a thinking animation so you can't talk to them. I don't have much information about becoming a best friend to an animal. As it appears it depends on how much you interact with them while they live in your town.
For example, Pippy was one of my original villagers. She ended up becoming my best friend when she escorted me to my birthday party. So for her case, it made sense she was my best friend. I had her for so long, and spent a lot of time with her.
On the subject of being best friends with your villagers, I recommend going above and beyond for their requests. Like, if they ask for a random ocean/river fish, give them the most expensive one available, if they ask for a fruit, give them perfect fruit instead, if they ask for furniture, get them something rare like a customized item or a spotlight item... Those sorts of things make them really happy, and you'll become friends faster.
Once Apple replied to me after I kept talking to her and in the middle of a sentence she said "no more!" because she said she thought she must be seeing clones of me. It was hilarious!
They'll begin to have a little thought bubble above their head, tell you that they're thinking about something, and that they'll talk later! So, not necessarily annoyed, but they do seem to get 'tired of it'.