The routes visible on komoot are public and rather easy to hike. Following other non-paved trails into the forest/jungle is only allowed with a ranger according to park regulations.
The park is easily worth 1-2 days visit, more if you book a guided tour. It has a primate rescue center, turtle rescue center, botanic garden with other (caged) animals and a cuc phuong museum. You can visit several caves and old trees on your own. You can rent a decent mountain bikes at the entrance. The main street can be driven by bycicle, motorbike or car.
To go deeper into the forest, ask for guided tours (you can camp overnight in the jungle!). There are not so many animals, but they are best seen in the early morning or late afternoon.