What's the best (cheapest and fastest) opportunity to get from Skövde to Jönköping by public transport during a weekend?
Skövde to Jönköping on Friday evening and back on Sunday evening.
I'd just take a train. According to SJ's very functional site, there are about hourly departures between the cities, with the last one from Skövde C on Friday at 20:54. For return trip, last train leaves Jönköping C at 21:59 on Sunday.
2nd class ticket costs either 136 or 154 SEK (depending on if it's operated by Västtrafik or SJ). There are plenty of direct trains, but with some you'd have to change at Falköping. Travel time by train is between 1:05 and 1:14.
There are buses too: travel time 1:30; prices from 129 SEK. Last departure from Skövde resecentrum (= central train/bus station) on Fri at 18:55, and from Jönköping on Sun at 21:25.
There are two options - bus and train - that both cost about 200-300 SEK depending on when you book. For the upcoming weekend the bus price is 258 SEK. Check out swebus.se for details. With train you end up at 272 SEK. Check out sj.se for details. The train at 1''05 is 25 minutes faster than the bus.