It's been my experience that scratches, even when they break the skin, are seldom serious. You should treat them immediately with hydrogen peroxide and a topical antibiotic but otherwise shouldn't worry about them. And it sounds like you were scratched, not bitten
Bites are a serious matter. Cats have powerful jaws and by biting can inflict deep puncture wounds--even kittens can break the skin. Cats have mouths full of bacteria--worse than dogs--so bite wounds do tend to get infected.
A cat's incisors (fangs) are long and smooth; they will penetrate skin and flesh quite easily. When dogs bite they also tend to shake or jerk their heads, resulting in a free-bleeding laceration. The bleeding actually helps cleanse the wound to a degree. Not so with cats. When the cat withdraws its teeth the wound may bleed a bit but will quickly close. The bacteria in the wound will multiply, causing it to fester. This is painful and often requires professional medical treatment with antibiotics.
Cat bites should be taken seriously, but at the same time one should not overreact. Unless you work full-time with stray and feral cats it is not worth it to get immunized for rabies--the disease has a very low incidence in the US cat population. On the other hand tetanus boosters are a good idea. Painful--but better than the alternative....