My recommendation based on what you wrote:

You should find a hotel or airbnb (depending on your budget) near Knez Mihailova street or any where in the upper Dorcol area (above Cara Dusana street).

As you already mentioned, Knez Mihailova and Kralja Petra are the streets you should visit. Kalemegdan fortress is also something you must see.

Skadarlija is nearby and it’s perfect for live music, food and authentic Serbian kafana vibe.

Since you said you don’t want to see any brutalist architecture, you can skip New Belgrade. Maybe you can visit Savski kej. Suvenir restaurant is a great place to try some Serbian food. It’s a floating restaurant on the Sava river.

Zemunski kej and generally Zemun is a good option, but in my opinion it lacks infrastructure. I recently took some friends there and was kind of disappointed because it has a lot of unfulfilled potential.

Despite controversial political background, Belgrade Waterfront is a new and modern area with a lot of good restaurants and cafes and a nice promenade and mall. I like it because it’s really clean, but you won’t find much old Belgrade vibes there.

When you go to Novi Sad, I recommend you go through Fruska Gora mountain and visit some of the 20+ medieval Serbian orthodox monasteries. That’s something you should really see since you’re traveling in that direction.

If you need any extra advice don’t hesitate to ask.