score:5
Yes, "Köpenhamn Ingerslevsgade DGI" is Copenhagen (and Göteborg is of course Gothenburg).
Ingerslevsgade is where most long-distance buses from Copenhagen leave. It's near the central station, just around the corner from the outer end of the S-train platforms. There are frequent train services from the airport to the central station -- continue straight ahead after exiting the baggage reclaim and go down then ramps to Platform 2; all trains from this platform go to the central station. There are ticket machines near the top of the ramps. One trip from the airport to the central station costs 48 DKK.
750 SEK for a train Copenhagen-Gothenburg sounds very expensive. The standard price for an Øresundståg ticket (as shown on oresundstag.se) is 454 SEK. No prebooking required for the Øresundståg trains, which leave every hour during the day. (There are a few faster and more expensive express trains on this route too, but they probably won't concern you).
The relevant timetable for the Øresundståg is at http://www.oresundstag.se/globalassets/tidtabeller/goteborg_halmstad_helsingborg_malmo_kopenhamn.pdf. Note that even though this timetable shows the trains originating at København H (the central station), they actually come from Helsingør, so there are a few additional options for boarding the train in central Copenhagen. The train that is shown as departing from København H at xx:32 actually departs Østerport xx:21 and Nørreport xx:24.
The last train leaves Copenhagen at 20:32, so there should be time to do some exploration of Copenhagen if you so choose -- or take an earlier train and get to bed in Gothenburg earlier if you find on the day you're too tired.
Upvote:4
There's a train leaving from the Copenhagen Airport terminal at 13:38 next Thursday and you can book direct to Gothenburg. Travel time is about 3 hours.
Regarding the 'tips-and-tricks' part of your question... Alternatively, to take in some local culture, you can catch the train from Copenhagen to Helsingør and check out Kronborg Castle, the traditional setting for Shakespeare's Hamlet (and a great tour spot in its own right). From Helsingør, there's a passenger ferry across to Helsingborg that lets you off almost right at the train station. Trains from Helsingborg to Gothenburg are on Sweden's main trunk and there's regular departures. The travel time on this route (not counting tourist stops) is about 4 1/2 hours.
If you wanted to stay in central Copenhagen, there are trains leaving about every 40 minutes from the main station (24/7). The travel time is about 4 1/2 hours.
There is no particular need to book ahead of time; that decision lies in how you define 'major rise in prices' versus 'flexibility'. Buses are available on any of the above routes and their prices are approximately 20% of train prices.