Is there a way to find how far along I am/how close I am to getting the Curious, Inquisitive, and Socratic badges on biology.stackexchange.com?
Yes, there is an easy way. Log in to Stack Exchange Data Explorer, and find Badge progress for curious, inquisitive, and Socratic on Biology.. Enter your user number from this site, which you can find in your url slug, and run the query.
You will get the results for each badge, with yes meaning you have the badge, and a fraction showing how far you need to go to get the others, if you don't have them yet.