For Liseberg, head towards Loke, Jukebox, Kallerado and Balder, then towards Flume ride and Mechanica where there is another small escalator or set of stairs towards aerospin. Afterwards head along the path towards Helix and Atmosfear via screaming swing. If you like burgers have Max, in the building after helix. If you're an Italian fan, do Lisebergbanan, head into the family area and eat at the Italian. Generally Italians right before thrill rides apparently doesn't end well, but I've never had this problem, just in case, after lunch do the rides in the family area.
I've missed some rides on here, but I suggest just doing them on route. By the time you've done all these you should by about 2-4 hours into your day depending on queues. This allows you to do rerides for the rest of the day/night. I'm just going to warn you that the park was clear by around 3 when I went. Everything was walk on, not even full trains. This was the last weekend in August, but in the morning I'd suggest following this route to get everything done. For Teatime, there's an stunning restaurant under Lisebergbanans station, themed to trains, it was about 7:00 when I went here and I don't think too many people realise it's there, and it was one of the Nicest restaurants I've been to in the whole of the country. That should be everything for the park.
For the city, I suggest getting a team out to the coast, where you can get a boat out around the archipelago, and get off at some of the islands. Heading back into the city, there is a few art museums although I didn't go in any. It is a very nice city to walk around in the centre as well. I'll try look into the actual city itself and recall it later on because my mind has gone blank.