I am in the process of developing an Android App that will sync all of the users bank and credit card accounts to get the balance. The purpose is that I want to build a budget app and I am not sure how to go about coding the app to import the account information. Does anyone have any suggestions for me in my endeavors?

Usually stuff like this is done directly on bank contract as they do not provide public APIs for such stuff to avoid DDOS or other attacks on their services.
Your only option will be web scraping where you need to work out URLs that bank is using. Then build request based on data provided to get page content and se something like Jsoup to look for specific elements and extract data
Good luck with it because this is very slow and painful process...

and scraping bank websites is NOT a good idea, you're almost certainly in violating of the bank's TOS if you do it.
And of course you'd need to fool your customers to allow your blatantly insecure app to log in using their real login details for the bank's website, NOT a smart idea.