How can I get the exact URLs/requests that a program is trying to connect to? I currently use Little Snitch to control network traffic on my Macbook Pro, and see what my apps are connecting to.

Unfortunately, Little Snitch only shows you the hostname that an app is connecting to. Is there a tool which will show you the full URL of the requests that apps are making? Even better if you can view the headers.
The most full featured way to peek into web traffic is to install and configure your system to use a web proxy.

The mitmproxy proxy (and mitmdump tool) is mature, reliable and well documented so I would start there unless you have another that meets your specific needs.

Also, widely used, superbly documented and well regarded is Charles Proxy