The trichlor pucks add from 6 - 9 ppm CYA for every 10 ppm of chlorine. It's a slow process when you start from zero at the beginning of the season.

You can use dichlor powder which is stabilized chlorine (similar to trichlor but in powder/granular form). The problem with dichlor is that it may contain other stuff that we don't want. So can the trichlor pucks. The other day I saw some that contained 1% copper. We definitely don't want copper. If you have access to a Sam's Club, their Kem-Tek dichlor "shock" is good. You would put that in the pool according to label directions so you'd maintain 5 or 6 ppm FC. I do not know how much CYA dichlor adds, I bet the mods have that info handy though. Use dichlor until the CYA level is around 40 ppm (+ or - 10 ppm) and then switch to just bleach.

What do you mean you've been going through "quite a bit" of bleach? How much is that on a daily basis? How does your pool bottom and sides feel? Slick? Or squeaky clean?

Until you have enough CYA in the water the sun will break down the chlorine very quickly.