Given that the outcome of your job search to a large extent depends on luck, it makes sense to spread your bets over a couple of years. Sometimes people test the waters one year, and if a suitable job doesn't appear, wait another year and try again. I'd recommend against putting off graduation for more than a year though. When you are ready, you should graduate. There are lots of opportunities that you may not see until you are getting out.
It is perfectly fine to take a postdoc position for a year or two, or a research job at an industrial research lab (such as Google, DeepMind, Meta, OpenAI, Microsoft etc.), or a national lab such as Los Alamos or Sandia, or research institute such as Santa Fe Institute, SRI, ICSI, Salk, or ISI. In such jobs, you are doing serious research and building up your vita, and you can apply for an assistant professor position again next year or in a couple of years when a suitable job opens up. In areas such as theory, almost all assistant profs have had a year or two of postdoc or industry research experience, and it is becoming more common in AI as well.
You should only take on a job in a teaching school if you really enjoy teaching and want to do it for a living. It is very hard to get any research done, and it is very hard to move to a research university from a teaching job. But if you are happy with teaching, there are many such jobs available in nice places.
As an industry researcher, you may have access to excellent resources beyond those in the academia. For instance, large computational experiments may cost tens of thousands (upto millions), and industry usually provides such access easily. You also do not have to teach, and there are software developers to help develop systems and run experiments. You also may have access to real-world data and customers, which can be a tremendous source of inspiration for research questions. You will likely also create patent applications, and their cost (about $30k) is not a problem in industry.
On the other hand, in industry you also may need to adhere to constraints on what you can work on, and may need to spend some of your time creating products and customer applications that may never be published, or even utilized.
Unlike jobs in academia, industry jobs can open up at any time. It helps if you've had an internship or two during your grad student years---it is always easier to apply when they already know you, but also you have a better idea of what you are getting into, and how to look for opportunities and how to interview.
It goes without saying that you should not start an industry job before you are completely done with your PhD. Even if you have just a couple of weeks of work left, that work can easily stretch into several years, and you will not be happy during that time. You can certainly apply before you are done, but make sure your start date is after your graduation date.
Your PhD research may indeed be a technical starting point for a startup. The most important next step is to find someone who knows the business side of it, i.e. how to get funding, how to run the company, etc. And you need to be ready to pivot, i.e. change the product / market / technology if you find out that what you first had in mind may not be successful.
It is still good to know what kind of salaries to expect, and before you go to job market, you should find out the latest. The "Computing Research News" publishes a survey every year on academic salaries in CS (the "Taulbee survey"). There are similar surveys on industry salaries, but they are much harder to access because in principle such information is not public. The best source is to ask recent graduates. Note that industry compensation is more complex because the salary is usually only a part of it (at the PhD level at least): there may be signing bonuses, annual bonuses, and stock options. Usually you need to compare offers in two dimensions: cash and stock.
Most academic openings in CS are announced in the October through March issues of the CACM, Chronicle of Higher Education, and CRA, but you can always send an application to a school you are interested in even if you haven't seen an announcement. For cognitive science or computational neuroscience jobs, you may want to check the society websites and mailing lists, although often those departments that are accessible to a computational modeler have already advertised in the CS-friendly mailing lists.
Many people apply to a lot of schools (i.e. 20 or more). If you don't apply you won't get a job. Previously it was easy when the applications were on paper, and maybe in the future it will again be when they may be in the same central database (as in some other more computationally savvy disciplines). Currently it is a lot of work because you have to fill in almost all applications online on separate websites (and your letter writers also have to submit them separately). So you need to be somewhat selective on where you apply. It is perfectly ok to contact someone in the school that you are interested and inquire whether they are hiring in your area. Often departments have very generic ads in the CACM and CRA list even though they have priorities for certain areas. On the other hand, if the school has advertised in the relevant newsgroups and mailing lists, they are probably seriously hiring in that area. For ideas about where you might apply, see e.g. the US News and World report rankings (there are many others as well, including global ones---such rankings are always somewhat controversial).
An application usually consists of a short cover letter, CV, a research statement, perhaps a teaching statement, and a few sample publications. The cover letter should just say that you are applying for an assistant professor position (or whatever). The research statement should describe your overall research philosophy and (especially) your research plans for the next several years. Similarly, the teaching statement (if any) should review which standard courses you would like to teach, and which new courses you would like to put together--as well as some teaching philosophy (e.g. about interactions, hands-on exercises, theory and practice, etc.). The vita should be as complete as possible, describe your research in one paragraph, and it should list your references. Get a hold of a few examples from successful people from recent years.
Most of the time the members of your dissertation committee write the letters, but if you know well-known people elsewhere who like your work, you should ask them too. You need three to six letters. Many schools ask the letters of reference to be submitted at the same time (or soon after) the application, and the rest request them from the references directly if you make the first cut.
Once you get offers you usually have a few weeks to decide, although that time is flexible. At that point you should contact all the other school that you might consider and ask whether there's a chance they'll interview you; usually they either invite you right away or tell you to take the other job. That way you can make a decision without surprises. As always, if you have at least two offers, you are in a better negotiating position. It can also be painful because many jobs can be attractive, and there are often many dimensions to the decision (ranking, interdisciplinarity, location, compensation, teaching opportunities, research support, etc.). One strategy is to decide what matters the most to you and make the decision primarily based on that dimension. Another one is to flip a coin and observe whether you are happy or disappointed in the way it came out :-).
Again, make sure you can finish your dissertation before you start your job, otherwise you are likely to get into trouble, or at least not be happy. Many candidates these days defer the start day by six months or a year, but often that is so that they can finish a project after their dissertation, or e.g. an appointment in industry. But don't even think about becoming an assistant professor and trying to finish your dissertation at the same time. It is not an option.
There are again lots of guides on how to be succesful as an assistant professor. But you also need to take into account the actual circumstances of where you are. Therefore, it is most important to identify a senior member in your department who can serve as your mentor. There's still a lot to learn...which of course makes it interesting and fun!