To be honest, this is also still pretty generic and I am not sure if you even had a look at the website I shared, which contains some of the main sites and other selections of places there. However, I'll try to share some activities that might not be too obvious and that I find pretty nice as an almost-local and would show someone:
- Take some time and have a coffee on Murinsel. Its a pretty unique
setting. Afterwards, you are able to walk down the river on water
level. All sounds of the city are gone, its an unique experience.
- Schlossberg is always recommended, if you walk up, do so via the square"Schloßbergplatz". You could get there by going through the tunnel in the mountain (starting next to the square "Karmeliterplatz" east), a pretty unique experience. There is an event venue inside the mountain, check if there if something of interest when you are there
- When you are on Schlossberg, visit the gardens in the south, Uhrturm and go further up afterwards. Next to the funicular railway station is a beautiful Biergarten. In my opinion one of the most underrated spots in Graz. There is also a restaurant nearby, but I don't know anything about its quality. "Kasematten" are also nearby, a former armory, then a prison, now a concert venue. Check for events on the days you are there.
- Hike down Schloßberg on its "backside", choosing one of the small paths north-west. This is definitely of the beaten path and gives you a great view along the "skyline" of Graz
Hope this helps you - please come up with more specific questions, I am happy to support. It's not that common, that someone asks about my almost home town on these forums ;-)