Ok, so heres a question- Are they purely spirits, or do they have an actual human identity as well? I mean, do they recall being spirits, etc?
The Wizards (Istari) are Ainu of Maiar (equivalent of lesser angels) rank. The Ainu are spirits that Eru Iluvatar (God in Tolkien's legendarium) created. The very first beings. After the creation of Arda the Valar (most powerful of the Ainu, equivalent of the Archangels) descended into Arda to shape it into something habitable for the coming of the Children of Iluvatar (elves, men; dwarves were regarded as stepchildren of Eru). The Valar and Maiar could thus take on the whatever form they deemed fit for whatever purpose. The likes of the Valar could even take on the forms of mountains etc.
However, if a body which they took was destroyed it would become increasingly difficult to create a new one to house themselves as Sauron eventually found out. In some cases they were bound to certain bodies by Eru until those bodies were destroyed (Melkor in his Dark Lord Visage for example).
The Wizard's 'old man' raiment was chosen for them by the Valar. The point was to restrict their powers and for them to not appear to powerful to the inhabitants of Middle-earth in the Third Age. They were sages, guides sent to aid the free peoples of Middle-earth. Not warriors sent to take on Sauron directly with brute force.
Sauron on the other hand was never restricted in this regard and his original raiment was of a very beautiful form. At the height of his power with his original body that he had crafted for himself, which was capable of shape shifting (snake, werewolf, bat) and with the One Ring, he was more powerful than his master (Melkor) at his lowest at the end of the First Age. That in itself is quite something as Melkor was second only to god at one point in time.
The Valar were far beyond the Maiar in terms of inherent might and power. It is said that even the most powerful of the Maiar (including Gandalf and Eönwë) would have succumbed to the will of the One Ring before the Crack of Doom just as Frodo did and Isildur before him and the elves would have shared the same fate. The Valar would have had no such problem. The One Ring would have no effect on them, as it was created by a lesser being than them.
That's one reason why some people theorise that Tom Bombadil may have been a Valar, not just a mere Maiar. But that's a whole nother complicated issue.
And to your final point, yes, they knew exactly who they were and what their purpose was. They knew their original names, which they used in Valinor and they most likely spoke to each other in Valarin, the language of the Valar. To quote Pengolodh "The making of a lambe [language] is the chief character of an Incarnate. The Valar, having arrayed them in this manner, would inevitably during their long sojourn in Arda have made a lambe for themselves"
The original names of the five wizards.
Saruman was known as Curumo.
Gandalf was known as Olorin.
Radagast was known as Aiwendil.
Alatar was known as Morinehtar.
Pallando was known as Rómestámo.
