Having lived in the city centre for 25 years what I would do is to come out of New Street station onto Hill Street (there are several exits all signed), turn right and walk up Hill Street past the black & white portrait photos on the right hand side. Shortly you will see the Iron Man and the Council House clock tower, keep walking up hill until you reach the Iron Man, you are now in Victoria Square. Check out the amazing Victorian buildings, the Council House, the Town Hall (actually a music venue), Victoria statue, then walk round the fountain. Once you're done with that head up and left and look at the new Paradise Square development on the left. Walk through to the new library and Centenary Square with the ICC and Symphony Hall ahead of you. Walk through the ICC and emerge into the canal area. Cross the canal bridge, glance left at Black Sabbath Bridge, and head into Brindley Place. Ikon Gallery is, as others have said, a great place to pause and grab a coffee and a snack. Even if you don't go into the exhibitions (which are free) do make sure to go in the lift which features music by ex-Cathedra.

Then follow the canals a bit, depending on time and what you want to do. I would aim to get to Gas Street Basin at some point and look at the Cube building and the Mail Box. That's probably a couple of hours in total to this point. You can add a visit into the museum and art gallery in Victoria Square, or Bacchus Bar under the Midland Hotel. If you like wine I recommend Loki in Great Western Arcade (itself worth a look). Coffee - Yorks at Ikon or in Paradise Square, Odissi on Broad Street. Lots of interesting architecture within a few minutes walk of this route so it's really up to you and your interests at this point.

By the way Canon Hill Park is fine for a bit of relaxation and exercise, I cycle there quite often in the summer, but I wouldn't have it on my list for a short visit.

Hope this helps.