"Then the claim would be immediate by
semisimplicity if one can show that
every irreducible representation of
$G(\mathbb{C})$ (or perhaps of Lie
groups in some more general class than
these) occurs as a subquotient of a
tensor power of a faithful one. How
might one prove the latter? (Can one
prove the latter for compact real
groups in a manner similar to the
proof for finite groups, and then pass
to semisimple complex groups by the
unitary trick?)"
Yes. The analysis is not so pretty, but it is elementary. Let $K$ be a compact Lie group, $V$ a faithful representation, and $W$ any other representation. Just as in the finite group case, $\mathrm{Hom}_K(W,V^{\otimes N}) \cong (W^* \otimes V^{\otimes N})^K$, and the dimension of the latter is $\int_K \overline{\chi_W} \cdot \chi_V^{N}$, where $\chi_V$ and $\chi_W$ are the characters of $V$ and $W$, and the integral is with respect to Haar measure. Let $d_V$ and $d_W$ be the dimensions of $V$ and $W$.
We now come to a technical nuisance. Let $Z$ be those elements of $K$ which are diagonal scalars in their action on $V$; this is a closed subgroup of $S^1$. For $g$ not in $Z$, we have $|\chi_V(g)| < d_V$. We first present the proof in the setting that $Z = \{ e \}$.
Choose a neighborhood $U$ of $\{ e \}$ small enough to be identified with an open disc in $\mathbb{R}^{\dim K}$. On $U$, we have the Taylor expansion $\chi_V(g) = d_V \exp(- Q(g-e) + O(g-e)^3)$, where $Q$ is a positive definite quadratic form; we also have $\chi_W(g) = d_W + O(g-e)$. Manipulating $\int_U \overline{\chi_W} \chi_V^N$ should give you
$$\frac{d_W \pi^{\dim K/2}}{\det Q} \cdot d_V^N \cdot N^{-\dim K/2}(1+O(N^{-1/2}))$$
Meanwhile, there is some $D<d_V$ such that $|\chi_V(g)| < D$ for $g \in K \setminus U$.
So the integral of $\overline{\chi_W} \chi_V^N$ over $K \setminus U$ is $O(D^N)$, which is dominated by the $d_V^N$ term in the $U$ integral.
We deduce that, unless $d_W=0$, we have $\mathrm{Hom}_K(W, V^{\otimes N})$ nonzero for $N$ sufficiently large.
If $Z$ is greater than $\{e \}$, then we can decompose $W$ into $Z$-isotypic pieces. Let $\tau$ be the identity character of the scalar diagonal matrices, and let the action of $Z$ on $W$ be by $\tau^k$. (If $\tau$ is finite, then $k$ is only defined modulo $|Z|$; just fix some choice of $k$). Then we want to consider maps from $W$ to $V_N:=V^{k+Nd_V} (\det \ )^{-N}$. $V_N$ is constructed so that $\overline{\chi_W} \chi_{V_N}$ is identically $d_W$ on $Z$; one then uses the above argument with a neighborhood of $Z$ replacing a neighborhood of the origin.
In the structure theory of an operator acting on an infinite-dimensional vector space over a field $F$, there is some good news and a lot of bad news. The best news (as Matthew Emerton mentions) is the classification of finitely generated modules over a PID. This beautiful theorem immediately generalizes the Jordan canonical form theorem and the classification of finite or finitely generated abelian groups. It says that every f.g. module over a PID $R$ is a direct sum of cyclic modules $R/d$, and the summands are unique in either the primary decomposition (which comes from the proof by isotypic submodules and Jordan blocks) or the invariant factor decomposition (which comes from the Smith normal form proof). You can take $R = \mathbb{Z}$, or $R = F[x]$ for a field $F$, or of course there are other choices.
I suspect that the bad news is likely to be uniformly bad for countably generated modules over any PID. I found a reference for the case of torsion-free abelian groups, so I will concentrate on that case first. Consider the abelian groups $A$ that lie between $\mathbb{Z}^2$ and $\mathbb{Z}[\frac1p]^2$ for a prime $p$. Those choices for $A$ that are free abelian are just lattices in the plane (whose coordinates have denominators that are powers of $p$). There is an important theorem from the theory of affine buildings that you can make an infinite tree of valence $p+1$ out of these lattices. Two lattices are equivalent if they are the same lattice up to homothetic expansion (by a power of $p$). Two lattices are connected by an edge if they are nested and their quotient is $\mathbb{Z}/p$. So, the theorem says that this graph whose vertices are lattices is acyclic and has degree $p+1$ everywhere; the graph is an affine Bruhat-Tits building of type $A_1$.
Given a general $\mathbb{Z}^2 \subseteq A \subseteq \mathbb{Z}[\frac1p]^2$, you can let $A_n = (\frac1{p^n}\mathbb{Z}^2) \cap A$. This is the type of direct limit that Matthew Emerton describes, which looks good, but bad news is coming. For a while, $A_n/A_{n-1}$ can be $(\mathbb{Z}/p)^2$, then it can be $\mathbb{Z}/p$, and then $A_n$ might stop growing. If it stops growing, then $A$ is free with two generators. If it always grows by $(\mathbb{Z}/p)^2$, then $A = \mathbb{Z}[\frac1p]^2$. But in the middle case, if it grows by $\mathbb{Z}/p$, then $A$ is described by a path in the affine building from the origin to infinity. There are uncountably many ($2^{\aleph_0}$) such paths, and they have a $p$-adic topology.
So when are two such modules $\mathbb{Z}^2 \subseteq A \subseteq \mathbb{Z}[\frac1p]^2$ and $\mathbb{Z}^2 \subseteq B \subseteq \mathbb{Z}[\frac1p]^2$ isomorphic? The outer module is obtained by tensoring either $A$ or $B$ with $\mathbb{Z}[\frac1p]$, so the question is when $A$ and $B$ are equivalent under the action of $\text{GL}(2,\mathbb{Z}[\frac1p])$, which is a countable group. After localizing at all of the other primes, this question was studied by Hjorth, Thomas, and other logicians and algebraists. As Brian Conrad suggests, they have several theorems that indicate that the answer is wild, similar in spirit to the classification of infinite graphs up to isomorphism.
Now suppose that $F$ is a field, most conveniently a countable or finite field, and suppose that $V$ is an $F[x]$-module lying between $F[x]^2$ and $F[x,x^{-1}]^2$. Then you get the same geometric picture as before: The $F[x]$-free choices for $V$, up to homothety by powers of $x$, form an affine building which is an infinite tree, such that the neighbors of each vertex are a projective line $FP^1$. The complicated submodules are the ones that correspond to paths in this tree from the origin to infinity. If $F$ is finite or countable, then $\text{GL}(2,F[x,x^{-1}])$ is also countable, but the set of these paths is always uncountable and the group action is always far from transitive. I don't know if Hjorth's and Thomas' results extend to this case (because I am not qualified to read their papers), but my guess is that they would be comparably pessimistic.
There are other cases in which you get good news, basically by making infinite-dimensional operators look like finite-dimensional operators. The module $V$ in the previous paragraph has no $x$-eigenvectors even after passing to the algebraic closure $\bar{F}$. This is the same statement as that $V$ is torsion-free over $F[x]$. I think that a torsion module over a PID $R$ behaves much better, if it is the direct sum of its isotypic summands. For instance, if $x$ is locally nilpotent on $V$, then I think that it is a direct sum of Jordan blocks, possibly including infinite Jordan blocks (like the differentiation operator $x = \partial/\partial t$ on $\mathbb{Q}[t]$).
Best Answer
Speaking as a nonexpert, I'd emphasize that the subject as a whole is deep and difficult. Even leaving aside the recent developments for $p$-adic groups, the representation theory of semisimple Lie groups has been studied for generations in the spirit of harmonic analysis. So there is a lot of literature and a fair number of books (not all still in print). Having heard many of Harish-Chandra's lectures years ago, I know that the subject requires enormous dedication and plenty of background knowledge including classical special cases. Some books are certainly more accessible for self-study than others, but a lot depends on what you already know and what you think you want to learn.
Access to MathSciNet is helpful for tracking books and other literature, as well as some insightful reviews. Without attempting my own assessment, here are the most likely books to be aware of besides the corrected paperback reprint of Varadarajan's 1989 Cambridge book (I have the original but not the corrected printing, so don't know how many changes were made):
MR2426516 (2009f:22009), Faraut, Jacques (F-PARIS6-IMJ), Analysis on Lie groups. An introduction. Cambridge Studies in Advanced Mathematics, 110. Cambridge University Press, Cambridge, 2008.
MR1151617 (93f:22009), Howe, Roger (1-YALE); Tan, Eng-Chye (SGP-SING), Nonabelian harmonic analysis. Applications of SL(2,R). Universitext. Springer-Verlag, New York, 1992.
MR0498996 (58 #16978), Wallach, Nolan R., Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics, No. 19. Marcel Dekker, Inc., New York, 1973.
This old book by Wallach as well as another by him on Lie group representations are presumably out of print. In any case, textbooks at an introductory level which emphasize both Lie group representations and harmonic analysis (often in the direction of symmetric spaces) are relatively few and far between. That probably reflects the practical fact that graduate courses aren't often attempted and are inevitably rather advanced. On the other hand, there are some modern graduate-level texts on compact Lie groups and related harmonic analysis as well as books on Lie groups and their representations with less coverage of harmonic analysis and symmetric spaces.