In my opinion, Nakuru is a bit far for a day trip and I don't know if the wildlife you see in Nakuru is that much better than what you can see in Nairobi National Park. If you really want a traditional safari, Nairobi National Park is surprisingly good for being adjacent to a capital city. The last time I went, I saw a lot of the herbivores, a couple lions, and a cerval cat. We didn't see the rhinos last time, but they are commonly spotted. You can do Nairobi National Park in the early morning and then make your way to the David Sheldrick Elephant Orphanage and the Giraffe Center.
Otherwise, I highly recommend Lake Naivasha, which is a couple hours drive from Nairobi and you also drive through some of my favorite scenery in Kenya where you have a stunning view of the Rift Valley. In Naivasha, you can take a boat ride to see hippos and there are a lot of antelope, zebra, giraffes, and buffalo that frequent many of the lodges in the area. My recommendation would be to book lunch at Sanctuary Farm, which has fantastic food straight from the farm and has lots of animals walking around the property (where you are also welcome to walk around). Also, Hells Gate is a nice little park near Naivasha with the same animals mentioned above. You can rent a bike at the entrance and bike through the park to the gorge (8 km each direction) and can take a short hike through the gorge with a guide. If you don't want to bike, you can also take a car through the park, but I highly recommend going by foot or bike for at least part of the way as it's a completely different experience than when in your car.
I think Naivasha would be my recommendation as it will get you out of Nairobi and see more of the country, but you won't see any big cats or predators and it will be a lot of driving. With Nairobi traffic, you should definitely aim to have a very early morning start.
