Solving Convex Optimization Problem Used for High Quality Denoising
Boyd has A Matlab Solver for Large-Scale ℓ1-Regularized Least Squares Problems. The problem formulation in there is slightly different, but the method can be applied for the problem.

Classical majorization-minimization approach also works well. This corresponds to iteratively perform soft-thresholding (for TV, clipping).

The solutions can be seen from the links. However, there are many methods to minimize these functionals by the extensive use of optimisation literature.

PS: As mentioned in other comments, FISTA will work well. Another 'very fast' algorithm family is primal-dual algorithms. You can see the interesting paper of Chambolle for an example, however there are plethora of research papers on primal-dual methods for linear inverse problem formulations.