You are correct that the vast majority of AP's are on H1B. This is a 3 year visa, renewable once. This implies that you need to get a Green Card if you want to make use of tenure after six years. Most of us start the Green Card application fairly quickly as it takes at least a year, often several years, to process, even if it turns out we don't get tenure. (I believe you also must submit your initial Green Card application within 18 months of your job offer, too.)
The H1B can also take several months to process. Consequently if you're already in the United States on an F1 visa, it is a very good idea to apply for OPT for a one-year extension. Watch the deadlines for this. OPT is notionally a work-internship type plan, but has a quick turnaround. I was on OPT for my first year as an AP, based not so much on my job but on my F1 visa, and then in my first year immediately applied for H1B, and then quickly started the Green Card thing.
The standard provisions on spousal employment authorization usually apply. There are some exceptions. For example, Australian citizens have some special visa for easier labor access. Check if your country has this, but most of us are H1B.