Lots of ways to go through Downtown, depending on how comfortable you are with traffic. My bike commute involves going through NSide and Downtown, to destination on Second Ave across from SSide. (It'd differ from yours merely by adding one bridge crossing.)
I like crossing the Allegheny on the Fort Duquesne Bridge, rolling through Point State Park, Liberty, Stanwix, Fourth Ave, Grant St, Jail Trail.
Return: Jail Trail, Fort Pitt Blvd, Commonwealth Place, through PtStPk, back across FtDuqBr to NSide.
As to buses, at the moment it's $2/ride or $75/mo, though that'll change Jan. 1. Routes 51BDG will get you from SSide to Fourth Ave by Stanwix pretty quickly, bypassing a lot of the busy part of Downtown.
Figure on about 3 out of 4 buses have racks. It gets better every year as they retire or retrofit old rackless buses. Very often helpful, but just enough w/o to be irritating.
Where on NSide are you traveling to?