To Sweden:
The closest airport to Uppsala is Arlanda (ARN), which is Stockholm's airport as well.
Some low-cost companies fly to Skavsta or Västerås airports, which they list as "Stockholm airport". They are not. Skavsta is 180 km from Uppsala and Västerås about 60 km, but neither of them with good connections.
To Uppsala:
Arlanda is about 30 km from Uppsala, with easy train and bus links (about every half hour) to the center of Uppsala. The train takes about 20 minutes. The bus (number 801) quite longer, about 45 min. Alternatively a taxi costs 550 SEK (about 55 euros) and it takes about 30 min.
Please note: There are two train companies serving the route Arlanda-Uppsala. The SJ trains, from the (ex-)national train company, and the "Pendeltåg" (line 40) from the SL trains, which are the local trains serving the Stockholm region (metro-type. Timetable here in swedish). You need to get the right ticket for the right train before going down to the platforms.
