Hi Smokey
The "best" way from Nairobi to Arusha/Moshi is to go by aeroplane... but this depends on your budget
It is very easy to get from Nairobi to Arusha/Moshi by the Shuttle bus.
There are a couple running this service
Riverside Shuttle
Impala shutte
you can book when you arrive in Nairobi - generally with the hotel you are staying in. They leave twice daily - early morning and early afternoon.
To Arusha it takes about 6.5 hours.
Hope this helps