To get to Abisko, you have to first reach Kiruna. The fastest way to reach Kiruna from Stockholm is by taking the flight. If you book in advance, then you will be able to get a return flight from Stockholm to Kiruna for USD 110-120. But then, you really have to book them in advance, because during the peak season, flight prices rise pretty quickly.
Train is also available from Stockholm to Kiruna, but it takes almost 17 hours to reach. One way ticket costs 100 USD.
From Kiruna to Abisko, you can go by train. A 2nd class train ticket to Abisko will cost around 9-10 USD. You can book the train tickets at the official website of Swedish Railways
To reach from the Kiruna airport to the train station, you need to take a bus that will take you to the city bus station. It will cost you around 15 USD. From the city bus station, you have to take free shuttle bus to the train station. Even if all these sounds a bit daunting, it is actually not so. And you can get help easily from the Swedish people about directions. The Swedish are quite fluent in English and as such there will not be such language barrier.
You can also get a transfer from the Kiruna airport to Abisko by bus from Visit Abisko. The bus costs around 50-55 USD one-way. It is better to figure out how to get to Abisko from Kiruna beforehand because taxis are insanely expensive.