As I know ,the Sobolev space $H^k(\Omega)=W^{k,2}(\Omega) $ and $L^2(\Omega)$ are Hilbert space. So, they must have orthogonal basis. But I can't find it on my book. Where I can find it ? I
In fact , what I really care about is $H^1$ or $H^1_0$ . The symbol is according to Lawrence C Evans 'Partial differential equations'.
In general, you can take any linearly independent dense sequence and apply the Gram-Schmidt process to produce an orthonormal basis. The basis will depend very much on the domain $\Omega$ and you can't write explicitly an orthonormal system for all the possible domains. In most cases, it is enough to know that a basis with certain properties must exist and it isn't necessary (and usually impossible) to describe it much more explicitly than "we have performed Gram-Schmidt on a dense sequence" or "we have used the spectral theorem for self-adjoint operators", etc.
For $L^2(\Omega)$, when $\Omega$ is a closed interval or a product of closed intervals, you can use an appropriate trigonometric system and the corresponding expansion of a function in that basis is the Fourier series.
For $H^1(\Omega)$, when $\Omega = [-1,1]$, you can describe a basis of polynomials called Sobolev-Legendre polynomials by a recurrence relation or by a semi-explicit formula similar to the formula for the Chebyshev polynomials. You can read about it here.