As is already apparent from Steven Landsburg's answer, the question is not as elementary as one might think at first glance. Already for $K_1$ of curves, there are a couple of as yet unresolved problems. This is mainly due to the fact that the localization sequence relates the K-theory of curves over function fields to the K-theory of their (higher-dimensional) models.
Concerning (1):
Let me discuss a bit what is known about the structure of $K_1$ of curves.
Generalities: Some of the structure of $K_1$ becomes clearer when looking at motives or motivic cohomology. If $C$ is a smooth projective curve over a field $k$, then there is a splitting of the motive (Chow or Voevodsky, does not matter):
$$
M_k(C)\cong \mathbb{Z}_k\oplus \operatorname{Jac}(C/k)\oplus\mathbb{Z}_k(1)[2].
$$
Using Bott periodicity of algebraic K-theory, the two summands $\mathbb{Z}$ and $\mathbb{Z}(1)[2]$ each contribute one copy of $K_1(k)$ to $K_1(C)$. The remaining part of $K_1(C)$ comes from the motive $\operatorname{Jac}(C/k)$, which as the notation suggests comes from the Jacobian.
There are several ways to define groups related to this Jacobian part of $K_1(C)$. Rationally, we can use the Adams eigenspace decomposition
$$
K_1(C)_{\mathbb{Q}}\cong\bigoplus_{q\geq 0} H^{2q-1,q}(C,\mathbb{Q}).
$$
Known vanishing results for motivic cohomology, imply that the summands are trivial unless $q=1,2$. One copy of $K_1(k)$ is the weight one part ($q=1$), and the weight two part ($q=2$) is the combination of the other $K_1(k)$ and the stuff coming from the Jacobian motive.
There are integral ways of defining the weight two part: we have $H^{3,2}(C,\mathbb{Z})=CH^2(C,1)=\ker(K_1(C)\to K_1(k(C))$, and Bloch defined the group $V(C)=\ker(CH^2(C,1)\to K_1(k))$. This is therefore a group which encodes integrally the stuff in $K_1(C)$ coming from the Jacobian motive. This also is the part of $K_1(C)$ that is not yet understood, and subject to several conjectures.
[The situation of smooth affine curves can be dealt with using the localization sequence, reducing to the projective curves. The analoguous thing to $V(C)$ in the case of smooth affine curves is $SK_1(k[C])=\ker(\det:K_1(k[C])\to K_1(k))$, which alternatively is the abelianization of $SL_\infty(k[C])$. That this group can be rather non-trivial is in the answer of Steven Landsburg.]
Now I will outline some known results and open problems concerning the "Jacobian part" of $K_1$ of curves, the case distinction being in terms of the base field.
Finite fields: Everything is known in the case of smooth curves over finite fields, cf. Theorems VI.6.4 and VI.6.7 of Weibel's $K$-book. The "Jacobian part" is trivial, so that $K_1(C)\cong\mathbb{F}_q^\times\oplus\mathbb{F}_q^\times$ for $C$ a smooth projective curve over $\mathbb{F}_q$. The localization sequence allows to deal with arbitrary smooth curves over $\mathbb{F}_q$.
Global fields: For curves over number fields, there are conjectures of Vaserstein and Bloch for curves over number fields. Vaserstein's conjecture asks if $SK_1(C)$ is torsion if $C$ is a smooth affine curve, Bloch's conjecture asks if $V(C)$ is torsion for $C$ smooth projective. Both conjectures are equivalent by a result of Raskind:
- W. Raskind. On $K_1$ of curves over global fields. Math. Ann. 288 (1990) 179-193.
You can find statements of the conjectures, references to the relevant papers and the proof of the equivalence in that paper. These conjectures state that the part of $K_1(C)$ coming from the Jacobian is torsion (and hence trivial over the algebraic closure of a global fields). Though the function field case is not discussed in this paper, probably similar statements would be consequences of Parshin's conjecture.
Some further discussion on $K_1$ of elliptic curves over number fields can be found in the answers to this MO-question. As noted there, there is a recent paper of Kondo and Yasuda, in which they provide some computations of $K_1$ of elliptic curves over function fields. I think their computations would imply a complete calculation of $V(E)$ of an elliptic curve $E$ provided the Parshin conjecture is true.
Generic fields: Finally, the statements outlined above say that $K_1$ of curves over "small fields" can be understood and is not too big (although this depends on serious conjectures). In general, this would appear to fail rather badly. There is a preprint of Rosenschon and Srinivas in which they show that for $k\subsetneq K$ an extension of algebraically closed fields of characteristic $0$ and $C$ a sufficiently generic smooth projective curve over $K$, the product map from K-theory $k^\times\otimes\operatorname{Pic}(C)\to CH^2(C,1)$ is injective, so that the Jacobian part of $K_1(C)$ is quite huge.
Summing up, we are still quite far from actually understanding $K_1$ of curves. Nevertheless, we might hope that the structure of $K_1$ of curves can be considered an elementary question some day in the future.
Concerning (2): To also say something about your question (2), the tame symbol appears as a map in a colimit of localization sequences for K-theory. Take the curve $C$, finitely many points as closed subscheme $Z$ and the open complement $U$ - this yields a localization sequence. Taking the limit over larger and larger closed subschemes $Z$ gives a localization sequence connecting the K-theory of the curve $C$, the K-theory of its function field $k(C)$, and the direct sum of K-theories of the residue fields of the closed points of $C$. The tame symbol is then the boundary map $K_2(k(C))\to\bigoplus_{\kappa\in C^{(1)}} \kappa(x)^\times$.
