One false note that rang like a gong to me, however, was that when Data has a mental problem Geordi La Forge suggests he talk to "Deanna" about it, which sounded very wrong. I couldn't imagine La Forge calling her anything but "Counsellor" to her face and "Counsellor Troi" to anyone else, at least during the actual series. So it got me thinking about the tangled web of what everyone on the Enterprise calls each other.
Starting at the top, Captain Picard generally calls people by their rank-only during official situations, and then either their first names or Mr. Last Name (no Ms.!) if he's being what he would call informal. So in the ready room it's "Will", "Beverly", "Geordi", but that's as far as it goes. Even he always calls Deanna "Counsellor", which I like to think shows he's a little intimidated by her or has more of a sense of being in a continuous Doctor/Patient relationship than he does with Crusher.
On the second tier, you've got Riker, Crusher & Troi. They're all less formal people (Troi maybe professionally rather than by temperament), are on first names with each other and generally get there with underlings fairly quickly, unless they're mad at them. Looking up, they're all "Captain" or "Captain Picard", except Beverly calling him "Jean Luc" when they're alone. Actually, Beverly seems to have more variability in what she calls people based on circumstance than most; she's the only regular crewmember I can think of who calls Deanna just "Troi" to other people, which contributes to my feeling that they're not really that close, and just engage in the occasional girl talk because they're the only person each other has to do so without moving a long way across the ranks.
Side note: Pulaski would pretty much fit into this category, being somewhat of an iconoclast, except I don't think she'd ever Jean Luc the Captain, which again indicates this comes out of Beverly's sense of personal relationship than her sense of propriety.
Then we have the department heads: Data, Worf, & La Forge. They're more formal than the second tier, never calling the upper ranks by their first name and often calling lower ranked people by their last name rather than their first (Geordi calls Ashley Judd's character, whom he seems to work closely with, "Lefler" rather than "Robin"). Unsurprisingly, of the three La Forge seems the least formal personally, and so will sometimes go first names with someone he's trying to set at ease ("Reg" instead of "Barclay") or of course to make an unsuccessful attempt to date, but this is very much the exception. Also: Worf says Rank or Rank Last Name when referring to the other two, even when they're calling him just "Worf".
Side Note: I can't remember Yar's time all that well (and am not re-watching Season One anytime soon) but it seemed like she was also pretty formal, when not space-drunk.
We don't get much time at all with any lower officers actually talking; when we do they're very much either Rank or Rank Last Name. With one exception, who is the exception to every rule: Wesley Crusher. It seems like he's more likely to call people based on their personal relationship (like his Mom does, appropriately enough) but there's a sampling problem. He calls La Forge "Geordi", but unfortunately the other department heads are mononymical, so while he generally says "Data" and "Worf" rather than their ranks it's uncertain if he would use first names if they had them. I would guess yes & no, respectively. Looking farther up, it's "Commander", "Mom", "Counsellor" and "Captain Picard".
The message of all this? I guess just that the writers should (and apparently did) use name variations as a nice way to establish character.
Just for fun, contrasts:
Sisko, despite perhaps seeming less formal than Picard, is either Rank or Rank Last Name with just about everybody just about all the time, with the exception of Dax, who is Dax regardless of her first name (mixed, as a commenter points out, with "Old Man"). I would see him going to first names on retirement faster than Picard though.
The rest of the crew, perhaps reflecting the frontier nature of the post and the mixing of civilians & different services, are pretty much first names as soon as they get to know each other (perhaps faster than some would like, like Bashir moving to "Julian & Miles" with O'Brien). Even Kira & O'Brien, both perhaps the most formal and the most distant in rank, become Nerys & Miles once crazy plot developments make them have a child together. First names just with each other, of course; looking up, it's "Commander/Captain" to Sisko, except for "Benjamin" from Dax (which as with Crusher seems like a mixture of their past relationship with her inherent informality). Worf, the latecomer, never quite gets there with anyone but Dax (though I'm sure Bashir would have been thrilled to be called "Julian"). It's actually kind of funny (though plot appropriate) that Ezri is more successfully informal with people after a day than Worf was after three years.
I'm leaving out DS9 & Voyager's non-Starfleet people because that's extremely case-by-case.
Janeway is first names with anyone she knows well, and "Captain" to absolutely everyone except very occasionally Chakotay, perhaps reflecting that given their isolation she's sort of Queen & Mother to everyone as well as Captain, and so needs to be even more of an icon.
The counsellor being dead, the second tier is just Chakotay and the holodoctor. Chakotay is first names with everyone (present or not) except generally the Captain, perhaps reflecting his experience commanding the bridle-at-authority Maquis. The Holodoctor remains Rank or Rank Last Name with almost everyone, actually more formal than Data, though perhaps with more sense of saying "Commander" or "Lieutenant" in such way that it's obvious whether he likes you or not.
Tuvok is like an island unto himself, not on first names with anybody and even Janeway only occasionally calls him by his one name, despite their being each other's oldest friend on the crew (and therefore the oldest friend they'll encounter for seven years).
Torres, Kim & Paris are all first names with each other right away (except when Torres is playing power games with Paris, which is tragically something we don't see much of), but equally uniformly formal to those above them.
and finally TOS-
Kirk is last name or first name with just about everyone, reflecting his attempts at informalizing a very regimented situation but also that the show was much less concerned about specific ranks in those days. He also has a nickname for McCoy, something I don't think happens again in any Trek series until Neelix comes up with "Mr. Vulcan".
Spock & McCoy go "Jim" pretty early on, and use it much more consistently than Beverly uses "Jean Luc". Looking down there's not much consistency, though predictably Spock stays more detached than Bones does.
Uhura, Sulu, Scotty & Checkhov are pretty much last names with each other and their underlings, and Rank, Rank Last Name, or Mr. Last Name to those above them. Which, in a way shows that the first Enterprise, for all it's rambunctiousness, was the most formal as far as the lower ranks went.