You should let people know your version of Lightroom when asking questions here, just in case a methodology changes.
Having said that, Lightroom keeps all your photo development info in one catalog, and can import photos into different folders depending on your import settings: How to set file import preferences in Lightroom Classic CC This can be confusing at first. The important thing is that unless you specify a new catalog (an advanced thing to do) all your images will be kept in the same catalog..
There is no old or new set, there is just one expanding set. When you're in Library mode, all imported images should be visible in the left panel "folders".
It's often -not- a good idea to keep all your images in one folder. It usually makes more sense to keep them in folders arranged by date. You can use the collections section in the lefthand panel and create arrangements with wild abandon, noting that it is a "virtual" organizing scheme and that images will still remain in the same real folders. Click on the plus sign, create a collection, name it whatever you want, and drag images from folders into it. I repeat, the real folder structure does not change when you do this.
You can move images from one folder to another, but I'd recommend against it while getting your feet wet. Once you arrange images into collections (and after you have the hang of that, you can try collection sets) you can close the Folder window and forget about it awhile.
One other thing that often needs to be said is that Lightroom doesn't touch your image files; instead, all the development and collection changes you make are stored in the catalog, separate from your images.
That's sort of a beginners lesson and I hope it gets you on the right track.
I don't have any official training in lightroom so I apologize if my question is a rookie one.
After I import a bunch of photos into lightroom, what if I wanted to import more photos to that same group of photos? When I do decide to import more photos, in starts a new catalog(or whatever it's called) and clears the screen of all the previous photos, and then only the new imports are seen and editable. How do I make it so I can import more photos to the original bunch?
Also, after I do import new photos, where do the old ones go that were just there? And, how do I bring them back, or toggle between the new set and the old set?
What you are seeing is not a new catalog or anything like that. Lightroom has a few special collections, such as ‘All Photographs’ and ‘Previous Import’. They are in the upper left corner. Lightroom automatically selects the latter collection when you import new images (but you can change that in the preferences, depending on which version you use). That is why you only see the newly imported photos. You can see all photos by selecting the ‘All Photographsk collection, or selecting the folder they were stored in. I advise you to get a book about Lightroom to understand how it works.