I will not mention the standard tourist destinations like Kotor, Perast, Blue Cave, Budva, as sufficiently described in the internet.
Into the mountains:
The road behind Lavender Bay direction Niksic leads you into a different world with beautiful views of Kotor Bay on the way. You will find many restaurants with local farm food at reasonable prices and hiking opportunities.
Another option is the small and at times rough road going up below Lavender Bay past the car park,. This will eventually lead you to Herceg Novi. Ideal as well for morning runs, if you do not mind the steep incline, almost no traffic.
Nevidio Canyon:
2h drive close to the Durmitor National Park. Canyoning is a livetime experience with jumps between 3 m and 12 m (the higher onces can be avoided) not for the faint hearted.
We had excellent experience with below company, where the tours are conducted by professional rescuers, who have climbed most mountains above 7000 m in the world. The tour starts at the Eco Village Nevidio, which is as well a great spot for lunch after the canyoning.
https://canyonnevidioexplorers.com/
Kotor:
If you want to escape the tourist lines , try the ladder of Kotor to get a bird view of the city. the entrance is next to the Scurda canal outside of the old city.
On the way up there is a nice rest place called 'Cheese Shop' , prices a bit on the higher side, but considering the location not unreasonable.
If you continue the ascend behind the cheese shop and keep to the left you will enter a different beautiful universe after another approximately 1h climb with marked trails surrounded by trees and a cool breeze.
Tara Canyon:
Rafting is a great way to enjoy the scenery and get a cool refreshment (water temperature 4 - 10 degree depending on the season).
Our favorite provider is 'Rafting Tara River Canyon', slightly more expensive than some others, but well worth it with 2 excellent meals included. Day tours start at 10 am with breakfast, finish around 3 - 4 pm with late lunch.
Avoid weekends in the main season, over 500 boats on the river and jams at the place to access the river! Season is from May to September/October, conditons in May can still be challenging with the river flowing about 3x faster.
Durmitor Nationalpark:
Bobotov Kuk is a challenging, but beautiful climb and will take you 6 - 8 hours both ways if you start from the black lake parking lot.
Once facing the lake follow the path to the right along the shore until there is a path leading into the forest on the right. Once you see a wooden bridge cross the river next to the stone hut and follow the red arrows/red dots for Bobotov Kuk.