What nikki said. The canal route to Stratford is a day out on a mountain bike. Also, the Edgbaston Tunnel on the Birmingham & Worcester canal is closed for the installation of a sensible-width towpath, which rules out the non-confusing escape route from the Gastric Basin[1]. Prior experience suggests that non-locals trying to find the NCN5 route through the city centre (which isn't a terrible one) based on Sustrans' signage alone is a recipe for getting lost, even without the diggy hole that's going on around the Broad Street area. Study Openstreetmap carefully, use GPS or ask locals for directions to head towards New Street Station, and try to pick up the southbound NCN5 route from there.
Leaving Birmingham to the south without busy roads involves an annoying climb at some point, but once you're over that it's rolling lanes with a downhill bias most of the way to Stratford. I'd suggest following NCN5 to where it crosses the Pershore Road South by Kings Norton Park, turn left up the hill, cross the annoying roundabout, then follow Masshouse Lane / Primrose Hill up and out of the urban area. The NCN55 alternative takes you up a steep muddy bank.
The Good Bitâ„¢ is roughly Oldberrow->Morton Bagot->Spernal lane->Great Alne. Stop to admire the weir just after Great Alne, then head to Wilmcote[2]. Final approach to Stratford via The Ridgeway.
Joining those two up is pretty much left as an exercise for the reader, though I'd suggest avoiding the NCN55 route down Icknield Street (it's excessively undulatey and the surface is shite). With a loaded tourer I'd probably head for the *top* of Weatheroak Hill, then down Watery Lane towards Portway. Any route to Aspley Heath, Forde Hall lane (some climbing required, but it's the most conducive to carrying momentum, and has less motor traffic than the roads around Tanworth) to Ullenhall, then across to Oldberrow (the brief stretch of the A4189 is mostly downhill, and not too busy except during rush hour).
ETA: Apart from using the NCN55 canal bit to get to Primrose hill, this is what nikki's plotted above in her GPX above
[1] Just round the corner from the Sea Life centre.
[2] Avoids the steep climb at Billesley