You really need to get a vehicle and a driver who will look after you and take you to the lodges. Any tour company will help you, but I have used Bunson Travel in the past and for my last trip to Kenya this past January.
I personally know the owner and they are a very reliable company. They will recommend where to stay in the parks, you can stay in tented camps if you are on a budget, or go mad and stay with the Serena Hotel group which is very luxurious. Whatever you do DON'T go with Wago Wago Tour company run by Mr Njorgre. The man is a con artist.
Others are Somak and Rhino Tour companies. I've not used them but they still seem to be in business over the years so they must have good organisation.
I wouldn't go by public transport. They are not allowed into the parks and you cannot be dropped off at the gate and walk eg into Nakuru, it is miles, hot and has lions. You can check out my pages on Kenya and all the parks.
If you are going to Mombasa, please remember to take your anti malarials and sleep under nets, it's a high risk malaria area. You say backpacking. I would skip that idea and have a hard lockable case (Samsonite) to ensure that you have all your belongings. I lost a lot of clothes from my case when I stayed in Mombasa. Don't take anything you can't afford to lose or wear any kind of jewellery or fancy stuff. Keep your money passport and credit cards in a money belt around your waist at all times. Nairobi is notorious for robbery.
Don't worry about carrying your luggage, the driver will load and unload it for you.
Please check my pages for more detailed advice.
Kenya airways, the pride of Africa, as they call themselves is one of the three main airlines in Africa. The airline has good routes to sub-sahara countries from Nairobi. I have flown it twice from Lilongwe to Nairobi in a B777 and from Nairobi to Mombasa in an Embraer 170. The service is good and the crew are friendly.
The fleet has:
4 Boeing 777-200ER
5 Boeing 767-300ER
5 Boeing 737-800
4 Boeing 737-700
4 Boeing 737-300
3 Embraer 170 LR
Jomo Kenyatta International Airport is the main airport in Kenya. It's an old airport with two international terminals and one domestic terminal. The distance is 18 kilometres to the east of Nairobi.
Visa fee is 25 USD for a single visa. Children below 16 years are not charged visa fees.
Taxi from the airport to downtown has a fixed price of 1500 Ksh. You can also take the local bus (City Hoppa) for 50 Ksh (70 cent).
If it's your first time to Kenya and you are alone, it's best to take a taxi to the city. The fixed price is 1500 Ksh to the downtown (a little bit more than 20 USD). The distance is 18 km. The driver will take you direct to the hotel, and he will also help you to find different hotels if you have not booked in advance.
The local bus called City Hoppa (line no. 34) drive between the airport and the city for 50 Ksh. That is 30 times cheaper than a taxi. Take the bus if you have been in Nairobi before and know where to go off the bus. Try to avoid the high fixed taxi price of 1500 Ksh.
Unless you have a tour bus or a friend with a car, do take a reliable taxi to the International Airport. It's about 20 minutes out of the city centre. Beware when on- and off-loading. The airport drop-off zone is now safer than it used to be (prior to 9/11), but still... Inside the airport departure facilities are ok. Arrival is more chaotic. Some good souvenir shopping in the duty free shop inside, but not really a good place to sit doen and eat; limited to drinks and snacks. Same for the business lounges.
Finding a safe taxi isn't always easy in Nairobi. With JimCab you're ok. They are radio taxis and confirm whereabouts and destinations of each ride to a central taxi office. They operate on fixed fares and are very reliable. JIM means... "Just In a Minute".
Main office and taxi stands out in the Gigri area, UN avenue, and Village Market.