For example, I am writing about my protagonist and her long time best friend. How could I convey closeness between the two characters in a way that feels like it could be a real relationship?
When you've known someone for a long time, what happens?
You anticipate their needs and you know what their preferences are. You know what colors they don't like, that they use gloss instead of lipstick always, that they would never wear shoes with laces if they had a choice...sometimes silly things.
To convey that is harder without seeming trite. In older movies, a character knowing what another character drinks or takes in their coffee and tea was shorthand for this kind of thing. It was often trotted out as a way to show characters who haven't seen each other in a while having a previous connection. (And sometimes a way to show that they have changed in the interim "Still a bourbon rocks?" "No ma'm. These days it's a coke. No ice.")
What they might know about each other will depend on your world, how people interact, how old they are, and how long they've been friends. Knowing someone's favorite ice cream, while it could be a detail picked up in adulthood, is something that is more likely learned in childhood, and could be used to establish how long they have been friends, especially if it's an ice cream treat that is now hard to find.
This question is very very broad. You say "protagonist and her longtime best friend." The best friend could be a dude. Your protagonist could be 16 or 60, in any kind of culture.
So take what I have written and use the lens of whatever specificity you have for the situation--knowing how a character reacts after a break up, that they check their travel itinerary constantly before a trip, even though they've memorized everything anyway, that they don't like thick socks because they bunch up, which totally wouldn't happen if they didn't always get sneakers one size too big....what they can know will depend on the specifics of the characters and the world/culture in the story.
This is a pretty broad question and therefore this is a pretty broad answer. It all comes down to what is called the "telling detail". The word "telling" emphasises two things here. First it means "important", a detail that matters. Second it means a detail that gives information.
A warrior walks into camp. What are the telling details that let us know who he is and what his arrival means? It depends on the circumstances, of course. If he is limping, bandaged, and carrying a bloody sword in his hand, we know at once that he has come from battle and had been wounded in the fight. The limp, the bandage, and the bloody sword are the telling details that paint this picture for us. Much of the rest of his appearance we can fill in for ourselves, and if one reader imagines his tunic red and another imagines it brown, it does not make any difference to the story.
A man walks through a pair of swinging doors wearing a ten gallon hat, a six shooter, and a star on his chest. Fill in the rest of the details for yourself...
A gloved hand lifts a fine china tea cup with two dainty fingers. Zoom out. What do you see ...
A paddle glides through brown water glistening with sunlight while a harsh guttural squark breaks the deep silence. Zoom out...
If you see two best friends together, what are the telling details, the details that say instantly that they are best friends? Is it that they dress alike, or that they are constantly whispering, or that they bicker constantly but never seem to be apart? The details will be different for different friendships, but if you can find the two or three telling details (three is often the magic number in these kinds of things) that shout out to the world that these two are best friends, and a particular kind of best friends, then you have your answer.