Stockholm is a city of islands, and the island you pick sets the tone for the whole night. A first date in Vasastan feels nothing like a first date on Södermalm, even though they're a fifteen minute walk apart. Here's how I'd think about it.
Södermalm for the date that wants to feel a little bohemian
Söder is the obvious answer and there's a reason. The blocks around Nytorget and Skånegatan have the highest density of good small restaurants in the city, and you can move between them without committing to anything. Start with a glass of wine at Folkbaren or Babette on Roslagsgatan's southern cousin streets, drift toward Nytorget for dinner at Nytorget Urban Deli or Pelikan if you want something more old-school Swedish, and then walk west along Bondegatan until you hit Mosebacke.
The trick on Söder is to end the night at Monteliusvägen. The walking path along the cliff edge looks across Riddarfjärden to Kungsholmen and the City Hall, and on a clear evening it's the best free view in Stockholm. If the date is going well, this is where you find out.
Vasastan for the grown-up first date
If Söder feels too try-hard or too crowded with twenty-two year olds, walk yourself up to Vasastan. Odenplan and the streets around Rörstrandsgatan are quieter, more residential, and the restaurants take themselves a touch more seriously without being stiff. Tranan at Karlbergsvägen has been doing the same Swedish bistro food for decades and it still works for a date because the lighting is forgiving and the bar downstairs lets you extend the evening without making a plan.
Rörstrandsgatan itself is a four-block stretch of pavement tables in summer that's about as close as Stockholm gets to feeling like a small Italian town. Storstad and Lemuria are both reliable.
Gamla Stan if you want to walk
I'd argue against eating dinner in Gamla Stan because the food-to-price ratio is rough, but as a place to wander on a first date it's hard to beat. Start at Järntorget, get a coffee at Chaikhana or Grillska Husets Konditori on Stortorget, and then just walk. Köpmangatan, Prästgatan, the slope down to Skeppsbron. The point is to keep moving and let the conversation do the work. You can cross Strömbron to Kungsträdgården afterward if you want a proper drink.
Kungsholmen for the low-pressure version
Kungsholmen is where Stockholmers actually live, and a date here feels less performative. The walk along Norr Mälarstrand at sunset, with the City Hall behind you and Söder across the water, is the city at its most flattering. Mälarpaviljongen, the floating bar by Rålambshovsparken, runs from spring through early autumn and is one of the few outdoor spots that feels relaxed rather than scene-y.
For dinner, Lux on Lilla Essingen if you're trying to impress, or somewhere on Hantverkargatan if you're not.
Östermalm if the date is already going somewhere
Östermalm is expensive and it knows it. Save it for a second or third date, or for a first date where you've both already decided this is serious. Saluhall, the food market on Östermalmstorg, is back in its renovated home and it's a good afternoon move: oysters at Lisa Elmqvist, a glass of something cold, and then a walk through Humlegården or up Sturegatan.
Riche on Birger Jarlsgatan is the cliché answer for an Östermalm date and it's a cliché because it works. Get a table in the front room, not the back.
A note on logistics
Stockholm is small enough that you can change neighborhoods mid-date if the first move isn't working. The metro runs until around one on weekdays and all night on Friday and Saturday, and a Voi scooter will get you from Söder to Vasastan in about twelve minutes if you're feeling brave. Don't over-plan. The city is the kind of place where the second bar is usually better than the first one you booked.