If you believe you’ve been bitten and you’re concerned, the best thing to do is to consult a doctor, not this webpage. 🙂
Technically cicadas don’t bite or sting; they do however pierce and suck. They might try to pierce and suck you, but don’t worry, they aren’t Vampires nor are they malicious or angry — they’re just ignorant and think you’re a tree. Just remove the cicada from your person, and go about your business. Cicadas also have pointy feet, egg-laying parts (ovipositors), and other sharp parts that might feel like a bite.
Cicadas don’t have jaws (mandibles) like a wasp, mantis, or ant, built to tear and chew flesh. Cicadas don’t have stingers, like bees and wasps, meant to deploy venom and paralyze or otherwise harm their victim. See a video of a Japanese hornet to see what I mean.
Cicadas obtain sustenance by drinking tree fluids, which are relatively watery compared to human blood. Drinking human blood would probably kill a cicada.
Caution: Don’t hold cicadas in a closed fist — you can hurt the cicadas, and they might try to drink from your hand meat.
(Reference these meme groups for more info Entomemeology and Wild Green Memes For Ecological Fiends).
Actual photo. Even with an open palm, they might take a taste!

Here is a video of a cicada that has confused my thumb for a juicy tree limb:
Magicicada trying to take a drink from Cicada Mania on Vimeo.
See if you can spot the cicadas sucker in this illustration:

Here’s a photo of a cicada’s mouth parts:

There is also a chance that if you believe you’ve been bitten by a cicada, you might have been bitten by a Cicada Killer Wasp. The Cicada Killer Wasp is a large wasp that hunts cicadas and usually can be found around cicadas or often attached to a cicada. Cicada Killer Wasps normally avoid humans, but if you mess with one, it might attack.
Tip of the day: If you want to avoid cicadas, don’t use power tools, drills, saws, lawnmowers, weed whackers, leaf blowers, etc. in their presence. Cicadas think the sounds made by these tools and machines are other cicadas. Female cicadas want to mate with the male cicadas they think they’re hearing, and male cicadas want to compete. If you can, use these tools in the morning or close to dusk when the temperatures are cooler, and cicadas are less active.