Laundry shelves..... cheap as possible.... plywood on hand
Guess I'd try to use all plywood. Unless of course, part of the project is art or developing your skills or to make points with the other person that does laundry in this space (all good reasons to do more, sometimes!)
For starters, I'd cut a single shelf and load it with.... well, with 60 lbs. See if it sags. If it sags, try facing it with something or alternatively giving it a long support underneath instead of on the face. Whatever looks good to you and is cheap and takes away the sag. Then I'd do the rest like that.
Dados vs cleats..... both will take care of sheer (ie countering gravity pushing straight down). Tight dadoes will help reduce side-to-side racking, but then so will screwing it to the wall in the 4 back corners. Whether the screws alone will do enough to deal with the racking.... I don't know but my guess is you'll do well to add at least one tight strut at midheight and another at the top.
A bunch of the other commenters are way more experienced than me so think before you leap on what I say.