It obviously depends on what you want out of the T-111. I personally like Thompson's Stain and sealer, where it will still allow the wood grain to shine through, but gives the wood a nice color tint.
I would have used Hardipanel T-111 much easier to paint and no rot. Since you went with wood I'd use a opaque stain with a very heavy roller. You'll still need to brush the grooves unless they are very shallow. As to brands any premium would do. I tend to use Sherwin Williams for exterior paint. Rough sawn T-111 is a bear to paint. Been there, done that. Hardi gives you the look, but is smoother and takes paint beautifully.
My neighbor did a T-111 shed with Ben Moore oil based solid exterior to match the siding on his house. That was about 15 years ago and it still looks good.