I now understand the initial key from all the round keys will be the original 4x4 block for 128-bit keys, but I do not know how it would work for something else, like AES-256 or 192. Would it be that you copy the rest of the key and put it into the second block, but what would you do with the third and fourth columns? Thanks in advance!
For a graphical description of the algorithm Click here. Key expansion is described for 128 bits in this video. The algorithm's block size is same, 128-bits, for all key-lengths 128, 192, and 256. So for each round a round key of 128 bits is produced.
Key expansion is different for 128, 192 and 256 bits. It is described in section 5.2 Figure 11 of fips-197. Follow Appendix A.1, A.2 and A.3 of fips-197 document for key expansion vectors. The crucial part is that you need longer keys for key expansion of 192 and 256-bit versions of the algorithm.