My school can track my browsing when I'm at home and using the Google account they gave me, the laptop is my personal one. Is there a way that they could track what I do, even though I'm on my other Google account? My school email has been on my laptop for a year or 2 and I'm just concerned that they could track things such as my bank information and business information.
Google is quite capable of correlating multiple accounts from the same machine. I don't believe they make this cross-account information available to individual account holders but have not specifically checked for it.
The main point I wanted to make was:
"... Is there a way that they could track what I do, even though I'm on my other Google account. ..."
Don't log into Google except when necessary, such as to check email.
Google already has extensive tracking capabilities through a myriad of agreements with other web sites. Don't make things easier for them!
Probably not, unless you agreed to install some additional software. Elaborating on @schroeder's answer, additional software that somehow reports back to the school could in theory check your web cache or other artifacts on disk or see what is currently running on your device. Web browser extensions or plug-ins would have the easiest time seeing what you are doing in your browser outside of your school account.
To be cautious, you can use a separate account for your school stuff. However, if you installed something that has administrative privileges or was installed with administrative privileges, that could possibly have visibility into other accounts. So, to be the most paranoid, you could have a whole separate device.
Of course, that is likely overkill. Assess your risk (think about what could happen if they did see what you are doing and the chance that they actually could/would). If you are doing something illegal or that could expose you to large financial consequences (e.g., lawsuits or loss of a hard-to-replace job) or damaged valued personal relationships, then you may want to act paranoid, if you in fact installed additional software.