I am possibly Mumsnet's most slovenly poster. This does, however, mean that when I do clean, I need stuff that works! It's got to remove the past [mumble] months' worth of grime.
Vinegar is a good de-greaser, deodoriser (surprisingly) and anti-static. No good for bathrooms, though. 'Natural' chemicals like bicarb, acetic acid, borax, etc do work but they're still chemicals - and more expensive than your ordinary bottled stuff. I love bleach! Bleach is eco-friendly, despite the bad publicity, as it turns into salt & air when used with water. If you have a problem with the chlorine, pay a bit more for a peroxide (oxygen) bleach. You need loo cleaner for the loo, as bleach doesn't treat limescale.
If you're a nutter for naturals, the one & only range that works hard enough for me is - umm, that Australian stuff with a daft name. It's made from onion juice & lemons, iirc.
If I weren't a slut - and had a lovely new bathroom like you (congrats!) - I would use that water-repellent spray for shower screens; it works. Actually, I'd try it on the tiles, too - be handy if it keeps them cleaner longer :)