I've been stung a lot over the last 3 years as a beginning beekeeper... fortunately, just a little swelling and itching. I have noticed hardly anything happens on the fingers, soft tissue will swell more (lips, forearm, etc). I just scrap the stinger off, rub a little pepsodent baking soda type toothpaste on it & ice it later when I get home. If I was working the back yard hives I'd mix a little baking soda in water and apply instead of using the toothpaste. I know lots of people use different things, vinegar, copper penny, etc. This is just what works for me. The ice really helps if there's much swelling.
IF you get short of breath or swelling occurs away from the sting site or your blood pressure dumps (faint) CALL 9-1-1 THIS IS A LIFE THREATENING REACTION! I carry an epi-pen for emergencies when I'm doing presentations or workshops with other people in the bee yard. (retired firefighter/EMT).