So you've probably already heard of the famous airline Ryanair.
If not, then time to become its best friend. This is by far the cheapest airline with the most flights around Europe, but it's a gamble and there are also some minor inconveniences with this budget airline.
You are only allowed one bag--here's Ryanair policy
"strictly one item of cabin baggage per passenger (excluding infants)
weighing up to 10kg with maximum dimensions of 55cm x 40cm x 20cm is
permitted. (handbag, briefcase, laptop, shop purchases, camera etc.)
must be carried in your 1 permitted piece of cabin baggage.
Extra/oversized cabin baggage will be refused at the boarding gate,
or where available, placed in the hold of the aircraft for a fee of
£60/€60. If you are unsure, check at the Bag Drop desk before going
So make sure you are not carrying too much wait and either way you should be making short trips and trust me, packing really light makes life so much simpler.
The best Airport to fly from within Sweden in reference to Jönköping is Gothenburg City Airport (GSE). The one in Stockholm is much farther away and will cost you a lot in transportation.
The best way to get to the airport GSE is to take Swebus.se to the Nils Ericson Terminalen in Gothenburg (this and the GOT Gothenburg Ladvetter Airport are the only stops in Gothenburg). from there you will need to take a 30 min bus to the GSE Airport through flygbussarna.se
I would try and book as many trips as possible as early as possible to the best of your ability knowing your schedule, but don't neglect school all too much.
Other cheap airlines within EU are:
Wizzair.com (probably the next best)