I work in a school district. If I bring my laptop in and do some banking transactions will the administrators on the network be able to see my passwords and private information after I have disconnected from that network?

Possibly, depending on what monitoring they use.
In reality the Admins don't trawl through the logs on the off chance of finding this sort of information, unless they are instructed to do so.

As Ben said, it depends on how they monitor the system. But I can't imagine an IT person, especially in this job market, who would risk his job to do it.

Depending on what software the district has installed on your work laptop. First off I can not recommend online banking to a person using Windows unless they have taken the time to implement and understand and use security measures. I also would like to suggest that you have a personal computer and a work computer. I also work for a school district (in the IT department)keeping personal stuff off your work computer is a good idea for one reason. It is not your computer so you can expect no privacy for any data or information including browsing history and or key logs stored on the computer. If the computer belongs to the school they also have a legal right and in some cases a responsibility to if needed monitor or audit the computer and any information it contains.
A laptop issued to teachers is a powerful tool for your use in teaching, lesson planning,classroom management and student evaluations. Just remember that you do not own it and it's owner (the school district) has the right to monitor it's use.

If the school network channels all internet traffic through a gateway... that gateway is going to have records of every machine that access the 'net and everyplace they have been... and as others have mentioned... depending on what data they trap... it could be very inclusive.

What everyone else has said is completely valid...

First of course the issue of what blockers they have to stop access to some public sites on the system to start with.

Second what is the school policy on use of internet though school system. Even though it is YOUR computer, the rules of internet use still apply using their system.

Third, most school have log systems to show what computer is used and when, plus many have log systems that will store key info.

This of course does not mean anyone will use it since IT people with access to that, often have access too much higher levels of security for many things.