Of course Shwe Dagon Pagoda is THE real must-see spot in Yangon. Go late in the afternoon and stay past sundown when it is all lit up. Really awesome. If you have a tripod take it along for incredible night time photo ops.
Other than that frankly I didn't think there was all that much to see. Sule Pagoda is somewhat interesting, but there are a million temples and I found Sule somewhat underwhelming.
My last day in Yangon my guide took me on a ferry across the river to an area which is much poorer to see how more of the "real people" live. The area down by the river where the little shuttle boats come and go was interesting for a quick visit.
You shouldn't lose a day going to Bagan. Most of the flights are VERY early in the AM. So you will arrive in Bagan by around 9 AM and have a full day ahead of you. Likewise flights out of Bagan are usually fairly early as well.
You may want to do a day in Yangon, then 3 days or so Bagan, then maybe fly over to Inle Lake for a couple of days before returning to Yangon for a final day.