You can go to Nonthaburi both by bus or boat but I always like to travel by boat as there is a lot more nice view along the river. The Nonthaburi pier is the end of the boat line and right in front of the pier is the central of Nonthaburi town.
The Chao Phraya River Express Boat run daily from 6 a.m. to 6.30 p.m. between Wat Ratchasingkhon (near Krungthep Bridge) and Nonthaburi province ( ticket price = 4 - 16 baht but in rush hour can be up to 25 baht ).
you can jump on the cheap boat, at any of the stops along the chao phraya river(west side of Bangkok), for only 13 baht and ride all the way to anywhere you wanna go including nonthaburi, i believe you can also buy a day ticket if you wanna go wandering around the west and east sides of the river for cheap. You can hop some faster, but they will be a good bit more expensive. This is definitely not the fastest way to go but its leisurely. It runs from 6am - 6pm
I'll update this again with a picture or 2 and certain pricing and maybe a name of the company next time i go to it
The best way to visit Nonthaburi is via the Chao Phraya River Express Boat, Nonthaburi is the very last northern stop. The trip takes a little over an hour by the regular boat (12B) and 45 minutes by the express boat (18B).
You can get the ferry across to Ko Kret at the Wat Sanam Nuea pier Pakkret just pass the market .It cost only 3 baht for the normal ferry.It departs whenever they have enough passengers and the trip crosses the Chao Phraya is just 2 - 3 minutes . There are also the faster longtail boat if you don't want to wait for about 30 baht.
Catch the 367 on Tdiwanon road and you can get from Major Pakret all the way down to Major Rangsit and Future Park. I use this bus almost everyday from where i live, in the middle of Tdiwanon.
7 baht for most trips if you're only going halfway, all the way from rangsit to pakret is 11 baht