Basically you earn 1 "point" for each one you find. Those "points" are only good for bragging about having that many finds. There are digital badges you can earn that are a way of showing achievements. The objective is just to get out, explore your surroundings, and have fun.
As for internet connection, I had a pretty slow one for many years playing the game. While you are out searching for the Geocaches the only thing you need is a GPS or a smartphone with a Geocaching app on it. You can even download the data to the phone and use it without a data connection. If you have said smartphone you can log the finds in the field. If you use a separate GPS you can come home and use the computer to log the finds.