Do Financial Transactions Affect the GL Immediately?

We are wondering if the financial transactions we perform (such as investments, loans, or gifts) post to the General Ledger immediately, or do we have to run the Post Transactions report to have them show in the GL activity?

This depends entirely on how you have configured your system. For example, for donations, this is controlled on the General System Setup screen. Click the Financial tab, and look at the selection for "Post gifts to General Ledger":

For investments, loans, and other Financial Services transactions, this is controlled by the GL Posting Method selection on the Financial System Settings screen:

If you have selected "Online" on this screen, transactions are posted to the GL as soon as they are processed. If you have chosen Batch, the transactions will be reflected in the GL only after you run the Post Transactions process. And, if you have selected Queue, the transactions will only process when the queue processor is run.