Tire recommedations needed

I have a ' 74 Raleigh 3-speed. I want some tires with more thread
than the typical Walmart brand.

There is quite of bit of extra room as far as a tires get close to
touching the fender.

I currently have a 26x1 and 3 eights, a shop said I could go to a
26x1.5 but I did not want to pay $35.00.

I had a flat about 5 miles from my house today, so I am looking for
ideas for reducing that.
(Sidewalks are favorite places for bone-heads to throw their beer
bottles.)

Thanks,
Andy