This has never been adequately explained, but we can make some guesses.
Why was Revan on board the Endar Spire?
From the context, I think it's fair to say that the Jedi weren't transporting Revan anywhere; the Endar Spire was meant to be his assignment. Consider Trask's lines from the prologue:
Trask: We work opposite shifts, I guess that's why you haven't seen me before.
Trask: You swore an oath just like everyone else on this mission. Now it's time to make good on that oath.
Even if we assume that the Jedi Council didn't know about the bond between Revan and Bastila, it seems likely that Bastila's mission was to keep an eye on Revan, possibly looking for signs of his Sith Lord personality re-emerging, but we can't be sure.
As for why that ship in particular, that's unclear. As a tie-in to Knights of the Old Republic II, LucasArts posted an 11-part series of background articles called "Chronicles of the Old Republic". One of these articles suggests that the Endar Spire was part of a larger fleet, intending to liberate Taris. The originals have been removed, but GameSpy.com has a backup of them:
In the skies above the Outer Rim world of Taris, a Jedi battle fleet engages the forces of Darth Malak in a desperate effort to halt the Sith's galactic domination. A band of freedom fighters, including Bastilla Shan, CARTH ONASI, MISSION VAO, and the Wookiee ZAALBAR, commandeer the legendary smuggling ship the Ebon Hawk and make their way with the amnesiac to the Jedi Enclave on DANTOOINE.
However, the "Chronicles" have many inconsistencies with the facts of the game, and so should be taken with a grain of salt, although they are considered an official source.
Even if we take "Chronicles" at face value, there's no good explanation about why they needed Revan on that mission. It would be totally reasonable to have Bastila, with her Battle Meditation ability, at such a crucial battle, so it's possible Revan was just going where she went.
Why were Revan and Bastila so unprotected?
First off, nobody else on the Endar Spire knew who Revan really was. Carth certainly didn't, and neither did Trask. There's not a lot of sense in the Jedi Council revealing that information to anyone; even if word didn't somehow get back to Malak, Carth's behaviour when he finds out later in the game suggests that things would have been difficult for Revan is anyone had known.
Also, Bastila was not entirely unprotected; we see at least one other Jedi on board the Endar Spire, and Trask's dialogue suggests that there are others.
What was the Jedi Council's plan for Revan?
This is unclear, but for my money the most plausible theory is that they had no plan to train him as a Jedi, they just wanted to use his tactical skills.
Revan is frequently described as a master tactician, to the point that he basically won the Mandalorian Wars single-handedly; Canderous Ordo, one of your companions later in the game and a former Mandalorian soldier1, comments on this at one point:
Canderous: It was not [the Republic's] ships or your men or your vaunted 'fight for freedom' that won this, the final battle of the [Mandalorian Wars]. It was by the actions of one person–the Jedi Revan–that you prevailed. Revan's strategies and tactics defeated the best of us. Even Mandalore himself was taken aback by the ferocity, the tenacity, and the subtlety of Revan's plans.
Having someone like that on their side is a pretty big win for the Jedi.
Under this theory, Bastila would be keeping an eye on him to make sure his Sith Lord personality didn't re-emerge. Depending on whether or not the Jedi Council knew about their Force Bond, she may also have been meant to secretly probe his memories for information about Malak's "secret weapon" (the Star Forge, not that the Jedi knew about it at the time).
1 For the uninitiated, Mandalorians are (one of) the Star Wars universe's Klingon expys, a meritocratic Proud Warrior Race, and Mandalore is the best of them. It may not seem like it, but in context this quote is very high praise.