At the very least, I'd recommend you:

• gloves - because you'll get really close to that gross shit. You don't want to touch it.
• a sponge - it doesn't need to be new; your old kitchen sponge is enough, just don't use it again in the kitchen. Use the yellow side to spread the cleaning agent, and the green side to remove obnoxious grime stuck to something. (Do it gently, and only with a really old sponge, to avoid scratching the surface.)
• a bucket - mostly to mix some soap and water.
• a dry rag - mostly for finishing/drying. A cringey old shirt that you won't be using again is usually enough.
• toilet brush - don't use the sponge to clean inside the toilet bowl; you'll be spreading the bacteria from your shit and piss to the rest of the restroom.

Everyone has the cleaning agents that they swear upon, so look for something that works for you. For me it's

• alcohol vinegar - to get rid of that brown crust in the sink (water in my city is hard as a brick) and around the shower drain. I usually apply it, wait a few minutes, then use the sponge to scrub it a bit. Then I remove the vinegar with the rag.
• bleach - exclusively used inside the toilet bowl. I squish some bleach there, then scrub it with the toilet brush, then flush it off, making sure that there's no bleach behind.
• disinfecting agent - I squish a bit of that inside the toilet bowl and just leave it there. It smells good, and it gets rid of the bacteria.
• an ammonium-based cleaning agent - I squish it on obvious grime on the walls (except the above), then scrub it with the sponge.
• soap and water - to "wash" the walls with the sponge.
• plain water with some disinfecting agent - to rinse it. Then I just remove the excess water with the rag and let the restroom to dry naturally (with closed doors otherwise my cats will step on the bathroom, step outside, and now I got to clean the bathroom again plus the corridor and furniture).

Important detail: do not mix any two of the cleaning agents that I've mentioned. Specially not ammonium and bleach.

For reference, the disinfecting agent that I use is called "pinho sol", but I have no idea if it's sold outside Brazil. You probably have some similar product wherever you live.