What kind of visa can I acquire in this situation? Which steps can I now make to get a visa with which I can work full time in the USA?
Unfortunately, the only derived status the spouse of an H-1B can automatically acquire is an H-4, which allows one to live in the US but generally not to work. The exception to the last bit is that, if the H-1B's employer begins the process of sponsoring the H-1B for permanent residence the H-4 spouse can apply for employment authorization, but as a practical matter many employers will only do this when the H-1B is getting close to the 6 year limit that status allows.
Beyond this the only alternative is to be offered a job which allows you to independently obtain your own employment visa. This might need to be an employer-sponsored visa like the H-1B, though certain nationalities (Canadian, Mexican, Australian that I know of) can qualify for visas that are slightly less challenging to get. Most of the possibilities will require professional qualifications.