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Since the treads have different depth on the sides, I should go with solid wood tread (custom cut for each tread). Right?
Yes it will be easy.
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For risers, I will paint the white. Is poplar good choice for it?
Most finish carpenter recommend MDF for paint grade material. It has a bad rap but for painted modling, risers etc. It is offers a superior paintable surface.
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Any recommendation on where to get them? Big box stores or woodcraft, stair supply store?
It depends on what species of wood you are going with. If you are using oak just go down to HD or lowes and get a 12” board of oak and cut it to size then get a bullnose bit for your router and bullnose the front.