Safari is Kiswahili for "journey".
If you only have one day, Nairobi National Park is perfect. It’s very manageable in size. On public holidays and weekends, it can get a bit more crowded, but if you have a good car you can leave most foreign and local tourists behind you.
There are more national parks that can technically be done in 1 day, but as a self-drive safari I wouldn’t recommend it. There are tour companies who offer this, which would be a safer option as you’ll be leaving and coming back in the dark. Plus these drivers are experienced and familiar with Kenyan roads.
But these parks can definitely easily be reached and explored over a weekend and Nairobi is a great base from which to do that.
If you want to go longer than just a weekend you can do multiple parks. Then it’s probably a good idea to rent a car with a foldable tent on the roof.
The type of car depends on the duration of your trip, the weather, and if you plan on camping as well. The most comfortable would be to rent a 4x4 as the roads are going to be bumpy and possibly muddy. But it is also possible to go with a “normal” car if these are your circumstances:
I have done this with 4 passengers and luggage in Ol Pajeta and it was definitely possible. It’s definitely the most budget safari you can do, but you’ll go a bit slower and there might be points where you’ll have to turn around or reverse. As long as the roads are dry this shouldn’t be a problem.
A 4x4 would be more expensive, but much more comfortable. You can reach most or all areas of any park and the weather would be much less of an issue.
Early morning or early evening, but then you risk getting out at dark.
It’s amazing to go exploring all on your own and not have any other cars blocking your views or having a lot of white people in safari cars in your pictures. But there’s a reason why these people have paid a small fortune for a guided safari: the guide. The guide or driver is usually in direct contact with the park rangers, which means they know exactly where the lions have made their last kill and where the newborn rhino is grazing with its mom. These tourists will get what they paid for.
But you can also get what they paid for! After the entrance there’s probably a few directions you can choose from. A good strategy would be to wait for a full safari vehicle to pass and follow them.
Take the number of a ranger or the number to reach the main gate in case you get a flat or need help in general.
If you find a ranger(s) along the way, you can ask them where you can find certain animals that you want to see (don’t forget to tip them). Or even ask them to be your guide for a while if you’re heading in the direction they’re going in. We did this in Ol Pajeta and paid 1000KSh for an hour and we dropped the ranger at the spot he needed to be.