So you got an offer to be an assistant professor in the computer science department at Prestigious U. Congratulations!
Time to negotiate your offer with the chair. Don't be nervous. This shouldn't be adversarial. Both of you have the same goal in mind--for you to come to Prestigious and be successful.
Let's discuss the different aspects of each package.
Funds for supporting your research such as equipment, graduate student support, travel and postdocs. Here you should explain what you need to be successful. This will vary by subdiscipline, a systems researcher will need more equipment and students than a theorist. Keep in mind the university is giving you funds for 2-4 years to start your research, after which you are expected to fund your own research via grants.
I don't recommend taking on a postdoc right at the start of your first academic appointment. Postdocs require good mentoring while you need to spend the first year getting your research up and running. If you do ask for postdoc money, ask to have a flexible start time.
Many departments give course reductions to get your research going. I'd suggest asking to spend your first semester teaching a graduate topics course based on your thesis research to pick up some PhD students followed by a semester with no classes to get you research program going.
This includes actual salary, which is also the base for future raises, and summer salary in the first couple of years. Feel free to ask for more salary, but often these numbers are fixed for new assistant professors. There is more give if you take an academic job later in your career. You could also say something like, "Well if you can't give me more salary maybe you could give me another semester of grad student support?"
It seems 80% of the time, a job candidate has a partner that needs accommodating. Don't wait until the end of negotiations, bring it up early. The more time we have, the better we can help. Doesn't matter what job they want--we know people and we know people who know people.
Many schools won't hire you as an assistant professor if you haven't finished your thesis. Has to do with college rankings work. Don't worry--they will generally give you some other role with the same package until you finish. This might delay your tenure clock though.
Delayed start time
A January start is usually fine with good reason but if you weren't planning to start until the fall of 2022 why are you on the market this year? If you do get the department to hold a position for you, remember you are also making a commitment--this is not an opportunity to try again for something better.
You may not get all that you want after a negotiation--don't take it personally. You shouldn't necessarily choose the place that gives you the biggest package. It's far more important in the long run that you pick a place where you can best succeed both professionally and personally, and the package is just a small piece of that puzzle.