Where can I find the calculation of the cohomology rings of the classifying spaces $BU_n,~BO_n$ and $BO,~BU$? I took a class where extensive use was made of these cohomology rings, but I missed the lecture where the result was derived (so I know what it is, but I don't really understand how it is obtained).
From what I gather, the teacher used spectral sequences and induction on $n$ for $BU_n$, so I would like a proof using the spectral sequence of the appropriate fibration. If this exists somewhere as a guided exercise (or if someone can make one) that'd be even better!
On a related note, I have no idea what $BU$ (and $BO$) actually represent. They were defined in the lectures as the direct limit of the $BU_n$ where one can inject $BU_n$ into $BU_{n+1}$ via the map $P\mapsto \Bbb C\cdot e_0\oplus\sigma(P)$ where $\sigma$ shifts the basis vectors of $\Bbb C^{\infty}$ by one. Contrary to $BU_n$, I don't understand what $BU$ is made of, and what a maps $f:X\to BU$ classify.
Which class are you taking? This material is not easy. You should ask your professor to ask for a proof or some hints.
There is a "simple" proof not using spectral sequences at here and is quite readable. Notice there is an obvious mistake in the proof.
I hope David Speyer or someone else can give an answer on the spectral sequence part(which I do not know). The classifying space $BU$ is defined as the quotient of $EU$ by $U$, much as the definition of any other classifying spaces. Using homotopy fibre of the inclusion $BH\rightarrow BG$ we can show if two $H,G$ are homotopy equivalent, then their classifying space is wealy homotopically equivalent. So this reduced the case of $BGL(\mathbb{C})$ and $BGL(\mathbb{\mathbb{R}})$ to the case of $BU$ and $BO$.
Now consider the fibration $$U_{n-1}\rightarrow U_{n}\rightarrow \mathbb{S}^{2n-1}$$ where we consider $U_{n-1}$ give an orthonormal basis for $\mathbb{C}^{n-1}$. So its embedding in $U_{n}$ left us with the choice of last basis, which is in $\mathbb{S}(\mathbb{C}^{n})=\mathbb{S}^{2n-1}$. Now consider the Gysin sequence we have
$$H^{i}(BU_{n})\rightarrow H^{i+2n}(BU_{n})\rightarrow H^{i+2n}(BU_{n-1})\rightarrow H^{i+1}(BU_{n})..$$
Here the first arrow is cup product with the Euler class $c_{n}$ of $BU_{n}$. The second arrow is the cohomological pull back of the above fibration map. The third is integration on $\mathbb{S}^{2n-1}$. By spectral sequences one should be able to conclude that $BU_{n}$ in degrees less than $2n$ are all even, which pulls back to generators of $BU_{n-1}$. Alternatively we can conclude this by noticing $U_{n}\cong \Omega G_{n}(\mathbb{C}^{\infty})$, so $BU_{n}\cong G_{n}(\mathbb{C}^{\infty})$ can only have even degree generators.
This implies the odd cohomology of $BU_{n}$ vanishes. So by induction we have that $$H^{*}(BU_{n})\cong H^{*}(BU_{n-1})\otimes \mathbb{Z}[c_{n}]$$once we establish the cup product with $c_{n}$ is injective.
Sorry I forgot the "reference" request, I recommend you to read Chapter 3 and 4 of Hatcher. This material is covered in detail in page 444, book page 435.