There sohould be a list of Ktheory and Khomology groups for the the standard spaces, like circle, spheres, (noncommutative) tori, but despite I've googled for it, I have found nothing satisfying. Maybe someone can give a reference?

$\begingroup$ Perhaps you could specify which flavour Ktheory you are talking about topological, algebraic, analytic...? $\endgroup$– Mark GrantMar 23, 2011 at 15:23

1$\begingroup$ What one calls "standard spaces" depends very much on one's perspective. $\endgroup$– André HenriquesMar 23, 2011 at 17:10

$\begingroup$ The reduced $K$theory of a sphere is the same as the $K$theory of a point, except that all the degrees get shifted by the dimension of the sphere. The (unreduced) $K$theory of a sphere is the direct sum of its reduced Ktheory and of the $K$theory of a point. $\endgroup$– André HenriquesMar 23, 2011 at 17:12
2 Answers
For many basic examples, the usual tools of (co)homology theory work just fine (and very similarly) in Ktheory and Khomology.
Here's an example. How does one compute, say, the De Rham cohomology of $S^1$? There are lots of ways, but one way is to use the MayerVietoris sequence  the same thing works in Ktheory (and Khomology). Write $S^1 = U \cup V$ where $U$ is a small neighborhood of the upper half of the circle and $V$ is a small neighborhood of the lower half. The long exact sequence in Ktheory looks like:
$$\to K^0(U \cap V) \to K^0(U) \oplus K^0(V) \to K^0(S^1) \to$$ $$ K^1(U \cap V) \to K^1(U) \oplus K^1(V) \to K^1(S^1) \to$$
We have $K^0(point) = \mathbb{Z}$ and $K^1(point) = 0$, and it's not hard to calculate that the map $K^0(U \cap V) \to K^0(U) \oplus K^0(V)$ is the map $\mathbb{Z}^2 \to \mathbb{Z}^2$ given by $(x,y) \mapsto x  y$. So we get $K^0(S^1) = K^1(S^1) = \mathbb{Z}$.
For noncommutative spaces, there is still a version of the MayerVietoris sequence which helps with some computations: it uses a decomposition $A = I + J$ of a C* algebra as the sum of two closed ideals. Combined with equivariant Ktheory/Khomology (and Bott periodicity!) these sorts of computations are often fairly routine. The challenge, as usual, generally is in working with specific (co)cycles that have geometric content.

1$\begingroup$ All that said, I did manage to find a brief list of spaces with their Ktheory computed: planetmath.org/encyclopedia/ExamplesOfKTheoryGroups.html $\endgroup$ Mar 23, 2011 at 18:54
The Chern character in $K$homology gives that, for a finite CWcomplex $X$, up to torsion (i.e. after tensoring with $\mathbb{Q}$), $K_j(X)$ is isomorphic to $\bigoplus_{i=0}^\infty H_{j+2i}(X,\mathbb{Q})$ (standard homology groups with rational coefficients). If $\dim X=2$ (and if I remember correctly there is also something in dimension 3), then this isomorphism actually holds over $\mathbb{Z}$: see Michel Matthey, Mapping the homology of a group to the $K$theory of its $C^*$algebra. Illinois J. Math. 46 (2002), no. 3, 953–977.