I got stung by a bee 3 times because I was shaking the trees for money..

Medicine is only for sale in T&T Mart or any other expansion, Nookling Junction will never sell it. Nookling Junction will upgrade to T&T mart after 10 days, provided you've bought enough from them (I think 10,000 bells worth of items, but I don't remember).

Some villagers will give you medicine if you need it. Any villager with the "uchi" personality will give you the medicine. Only female villagers can have that personality. Only 22 of the over 200 villagers in the game have that personality (I think I read there were 313 villagers).

Bee stings are not permanent. They tend to last only a few hours at most, then you'll be good as new. Try saving and quitting, and then reload the game. You may be healthy. It may take an hour or so for the sting to heal.

If you speak to a villager once, they will comment on your injury. If you speak to the same villager again, they will act normally. No villager will remark on the same injury twice. (Please note, if you are stung and talk to a villager, then are healed, and then stung again your villagers will remark on your injury again.)

Catching bees is easy. Hold your net and push down on the d-pad. This makes it so when you push left or right on the d-pad you will automatically get your net. With your hands free, shake your trees. If a hive falls, run north/up and push right or left on the d-pad. You will spin around and face the swarm with the net. Click and hold the A button and wait for the swarm to get close. Release the A button at the right time and you've caught a rather expensive bug. The timing takes a bit to get used to, but it should only take a few tries at most to get it down. Alternately, you can run into a building or across the railroad tracks to Main Street. The bee swarm will despawn and you can collect their hive for an easy 500 bells.

Nookling Junction doesn't sell medicine. You need to upgrade to T&T Mart, then you can buy medicine for 400 bells. Also there is a certain type of female villager that will give you medicine if you talk to them. (I think it's the big sister personality) Also, if you save and quit, your bee sting will be healed without having to wait a day.

You can open your train station and there will be no bees

Okay so save and quit first. Then press A once you get to the ACNL page. Press continue and character name that was stung. When Isobel askes 'shall we get started' press I need help 1st. Press 'set date and time'. Set the date to the next day and the bee sting(s) will be gone!
Hope that helped you out!

Save ans quit then open the game again. There is no need to change the time/day/year