Elan Valley cycle trail; family friendly graded singletrack all the way from Rhayader up to Caban Coch dam. Very pretty with carved benches, picnic spots, spectacular dams, heather clad hills, and if you're lucky, water pouring over the top dam.
Ystwyth cycle trail. Less spectacular, but relatively hazard free, close to the river from Llanilar on, so good access for paddling or swimming, couple of short road sections. You can go as far as Tregaron on that, and there are a few places where you can divert to small villages / shops, Strata Florida abbey near Pontrhydfedigaid.
The woods above Devils bridge (The Arch) have a forestry track route that can be made into a loop that goes over the top, right through the windfarm (used to be a very attractive bit of moorland with bee orchids and and rare mosses) to the Wye near Sweetlamb, then goes back on a different set of tracks via the Dilyw (a small river) to return to 'The Arch' most of that is bike trailer country.
There's also the Hafod estate; forestry owned, very scenic, criss crossed by forestry roads and full of interest, with wooden bridges, picnic spots, places to walk, but that is more walking oriented so it would mean leaving the bikes at various points to see the waterfalls, caves, japanese bridge etc.
Hope that helps
