1999 Hyundai Elantra starting issues

The terminals on the battery posts have to be tight. You mentioned that they get loose really easily. That could easily be your problem. I am still baffled though, you said that the cables were recently replaced. Then, how come the connectors keep getting loose? Did you not tight them really well?

Let's deal with one problem at a time, as replacing the battery for your car is costly. Where I work, your battery is called 24F84 (Group size 24F, 84 months prorated warranty) at 67.99. If money is no object, by all means, go ahead and replace it. But, before you do that, I want you to go to an auto part store and ask the attendant for this item. It is a replacement battery terminal (you said the current terminals keep getting loose, no matter how hard you have tighten them). Replace the one that keeps giving you grief. It is made by Lynx part number 01045. Here is the picture.

Once installed, go back to the store and have your battery tested. If the battery is proved to have bad cell, then you have no choice but to replace. Some places will sell you a different group battery for you (example 75DT). Please refuse and ask for group size 24F. I don't know about Autozone, but Checkers, Kragen, O'Reilly have them.