Originally Posted by WNG

They are referring to 10-speed as in there are 10 cogs on the rear hub, that's 2x more than what's currently on your Takara.

Your cheapest route is to utilize mountain bike and hybrid components. any MTB brake lever for cantilevers will work with centerpulls. Don't use V-brake/linear brake levers.
I'd spread the triangle to 126mm to use a 7-spd freewheel (indexed). Buy a cheap SRAM or Shimano 7-spd Indexing rear derailleur. They do come with hangers built in.
Hook that up to a set of MTB/hybrid 7-spd shifters. Pick you poison here, grip shift, Shimano rapid fire, thumb, etc.
Mounted on a riser bar. Can also swap the stem to a 1" MTB stem for more rise.
And do change that seat. Your @ss will thank you. A comfy seat makes a world of difference.

BTW, are those 700C wheels? If so, mount the widest tires that will fit on those rims and clear the brakes. They will be more comfortable and less prone to flats and damage. Perfect for a townie bike.