Definitely have to look at the Sudbury branch. Its diesel in electric land, the longest viaduct in East Anglia, the Chappel rail museum (with an excellent beer festival coming up) and Sudbury is pretty and you can walk or cycle along some of the remaining rail line to Cambridge
My thoughts exactly - and not just Chappel (although that is definitely worth the visit) also well worth a stop-off on the way at somewhere like Bures and a walk along the river. As noted, Sudbury is a nice town with good Gainsborough links. Rural north Essex/south Suffolk still feels a bit of a life-time away from the M25.
Another one to consider - the Harwich branch. Interesting start at Manningtree, with lovely views along the Stour estuary and then ending up (via Parkeston Quay - showing my age) and then into Harwich itself, which is a whole lot different type of harbour to Felixstowe across the way with the panamax ships. For stuff to see in Harwich etc, search Diamond Geezer's blog for his day out there (in fact, if you want quirky branch lines in the SE and things to see on them, he's probably been there and done all the necessary research for you. He did the Braintree branch/Silver End a few months ago).
And as noted, both the East Suffolk Line (get the bus from Halesworth to Southwold if it's still running) as well as all the Norfolk branches. If you want a real day of it, then early to Sheringham, and then the CoastalHopper bus all the way along to Lynn, getting off at whichever of the great beaches, small towns and Hun'ston as you wish.