Twante is the most accessible delta town from Yangon and makes for a nice half day or full day trip. A short ride on the Dallah ferry from the Pansodan Street Jetty on Strand Road (US$1, 10 minutes), and then a ride in a shared taxi (an automobile) -- or a bumpy 40 minute ride on a pickup (300 kyat) through leafy villages and large pools of water which gets you a good feel for life on the delta.
A slower way is by ferry from Yangon (2 hours, US$1) and, for a good look at delta life, consider taking the Twante ferry one way and the pickup/Dallah ferry on the way back.
Twante is renowned for its pottery and any trishaw driver will be happy to take you to the Oh-Bo Pottery sheds for a quick look (200-500 kyat).
Shwesandaw Pagoda, a short walk from the pickup stand, is a wonderful smaller copy of the Shwesdagon Pagoda without the crowds and well worth the trip by itself.