As you've seen, you can get very sick. While you may luck out and walk away from a rat bite or scratch with little more than a small, irritating wound, it's important to see a doctor right away.
There's a good chance you'll need to be tested for diseases and there's an even better chance you'll need antibiotics.
If you decide to take a "wait and see" approach you should head to the doctor immediately if you start experiencing joint pain, fever, chills, muscle pains, headaches, skin rashes, vomiting, or diarrhea. It's also a good idea to head to the doctor if you see pus at the skin wound, or any kind of skin rash. Be sure to keep the wound exceedingly clean.