Ezra - Lieutenant and/or Lieutenant Commander
Ezra is promoted early in the season 3 premiere "Steps into Shadow":
Commander Sato: Ezra Bridger, I trusted you with the last mission, and your success has not gone unnoticed. You will lead this mission as Lieutenant Commander.
Ezra: Lieutenant? A promotion?
Star Wars Rebels Season 3 Episode 1: "Steps into Shadow"
Ezra gets his command suspended as a result of his actions in this episode, but it's not clear what it does to his rank; in the episode "Twin Suns", he's referred to as "Lieutenant", which may or may not be a short form for the higher rank:
Rebel: Hey, Lieutenant. What you got there?
Ezra: Oh, some munitions for the A-wings. Where do you want 'em?
Star Wars Rebels Season 3 Episode 19: "Twin Suns"
However, in the season 3 finale, Ezra is referred to as a "Commander":
Sato: Commander Bridger, go to heading 221 and prepare to jump.
Star Wars Rebels Season 3 Episode 20: "Zero Hour: Part 1"
And in an early season 4 episode, Mon Mothma uses Ezra's full title, confirming that he is a Lieutenant Commander by this point.
Hera - Captain, later General
According to the trivia gallery for "Steps into Shadow" on StarWars.com, Hera is a full Captain in the Rebellion:
Hera now has a traditional rebel rank badge on her sleeve indicating she is a captain.
As of season 4, Hera appears to have made General:
Hera: Phoenix Two, you've got the wingman.
Mart: Already on it, General.
Star Wars: Rebels Season 4 Episode 9: "Rebel Assault"
And this is the rank she holds in Rogue One, according to Dave Filoni1:
"I can clarify that's absolutely Hera being paged, not her dad," Filoni says. "The movie gives that away, but I would rather have all the [references] be right. Hera will eventually become a general in the Rebel Alliance."
Kanan - Unknown, possibly Commander
There's a scene early in "The Holocrons of Fate" where a Rebel pilot addresses a Commander, with no name given:
Rebel Pilot: Coming up on the rendezvous point, Commander, but the transport's not responding. It looks like it was attacked.
Star Wars Rebels Season 3 Episode 2: "The Holocrons of Fate"
Since the only other people in the room are Kanan, Ezra, and another pilot, odds are good that he's referring to one of the Jedi. There's another moment in "The Last Battle" where Rex addresses Kanan as "Commander" before correcting himself:
Kanan: We should move.
Rex: Right Commander. Uh, Kanan.
Star Wars Rebels Season 3 Episode 5: "The Last Battle"
Upon first watching the episode, I assumed this was part of Rex flashing back to the Clone Wars; however, it does seem unusual for him to address Kanan as a "Commander" rather than a "General" if that were the case.
Sabine - unknown
Sabine has yet to be associated with a rank on the show. About all we can say is that she's subordinate to Commander Sato, which doesn't tell us a whole lot.
Zeb - Captain
The season 3 episode "Warhead" has Zeb frequently referred to as a Captain, by himself and others; a handful of examples:
Sabine: Carry on, Captain Orrelios.
EXD-9: [scanning Zeb] Species, Lasat. Designation, military. Rank, unknown.
Zeb: Captain Orrelios, at your service.
Rebel Officer [over PA system] Captain Orrelios to the command center for a priority message.
AP-5: Well, whatever the outcome, you performed admirably, Captain.
Star Wars Rebels Season 3 Episode 13: "Warhead"
Rex - Captain
Although this is also the rank Rex held in the Clone Wars, he's still occasionally referred to by this title; for example:
Hera: Sabine; I have a mission for you and Chop. Thanks to Captain Rex, I found an old base where we might be able to salvage some much-needed medical supplies.
Star Wars Rebels Season 2 Episode 3: "Always Two There Are"
Chopper and AP-5 - unknown; likely none
Droids aren't really considered people in the strictest sense, and there's no indication that they're ordinarily conferred a rank in the Rebellion, or in the Republic military on which the Rebellion is based.
There's an interesting quirk to that, which Donatello Swansino points out in comments, in K-3PO, the white 3PO unit seen on Hoth in Empire Strikes Back; K-3PO is seen with two red dots on its plating, consistent with the insignia for a rebel captain. However, I would suggest this is a special case, and not something we can apply to Chopper and AP-5, for a few reasons:
- Although Legends materials established K-3PO as having been granted the rank of Captain (likely to explain the markings), this has not been addressed in any Canon works
- Even in Legends, K-3PO being "recognized" was a long and difficult process, and one specifically mentioned as being unusual; K-3PO is the exception, rather than the rule
- Neither Chopper nor AP-5 have any exterior markings that could be mistaken for rank insignia
In all likelihood, the droids have no rank at all.
1 Hat tip to DBPriGuy for the source