I highly recommend the discussion in Shanahan's book, The Atiyah-Singer Index Theorem (An introduction), Lecture Notes in Math 638. In addition to a sketch of the proof, he gives a nice discussion of how the formidable general statement of the theorem gives the answers for your two examples, plus the (spin) Dirac operator and the signature operator. There are other treatments if you want to learn all of the details of the proof, but that book is excellent for the purpose of your question.
(Added later) A second reading of the question suggests that you are asking for a complete proof for these two cases, rather than instructions on how to deduce these cases from the full A-S theorem.
For (1), the ingredients are the Hodge theorem to identify the kernel and cokernel as the de Rham cohomology in even and odd dimensions. Then you need the de Rham theorem to identify these cohomology groups as (say) singular cohomology. This shows that the index is the Euler characteristic. Finally, you need to identify the Euler characteristic as the evaluation of the Euler class on the fundamental cycle of your manifold. You can find this latter in many places, eg Milnor-Stasheff.
I don't know that there is as direct a proof of (2), which is essentially the Hirzebruch-Riemann-Roch theorem.