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These sort of things. I've found them in dollar stores even, but they're pretty cheap from Amazon or Target. A quick Google for scouring pads should give you some sources convenient to wherever you are in the world.
If it's latex paint, warm it up. It will come up much easier. I'd plug the sink and put boiling water in there, let it sit a bit until you won't burn your fingers. Use a very rough plastic scouring pad and go with the slight grain of the stainless steel texture. That will encourage the softened paint to lift out of the slight textured recesses in one piece.
It's sticking fast because the sink is lightly textured. It's the same principle as lightly sanding something you are (intentionally!) painting if it is very slick to start with. Paint holds on to rough surfaces better.
Buckets, rubber gloves, something scrubby (these are what I think of first). It will take a lot of elbow grease. Start at the top of the tile and work top down.
Based on the pictures, I agree that the walls might be uncleanable, but go gently on them and it might be alright, although that isn't going to fix what is going on behind the wallpaper.
Barkeeper's Friend, a scouring pad, and elbow grease.
My mom was a cleaning lady. No one hates cleaning like the children of professional cleaners, because none of them can afford childcare, so when you're out of school for any reason they take you along and have you help, and help to professional standards. I dusted endless miles of baseboards in childhood.
This is a long winded way to say that I worried about my cleanliness too before my daughter was born.
- If you have it, throwing money at the problem is an option - hiring a weekly cleaning service would get you to a base level of cleanliness and keep you there. And as someone else suggested, you can watch what they do and copy that in between cleaning sessions. Don't be creepy about it and, like, stare.
- Pick your battles. No one will care (and more importantly, your child will not be damaged) if your blinds are dusty, or the laundry is in piles on the floor instead of in a hamper, or if there's some ketchup dripped on your refrigerator shelf, or the bathroom mirror is splattered with some toothpaste, or the canned goods from your last grocery run are still piled on the counter.
- Realistically speaking, the most important place to keep clean with a small child in the home is the kitchen, because they don't have the ability to fight off germs or deal with spoilage like you do yet, and they don't spend much time in the bathroom (the other germ-laden area of the house) until older except for baths. Keep reasonably well up on washing your dishes (or loading them into the dishwasher), though having some sitting in the bottom of your sink for most of a day won't hurt anything. Keep baby bottles and pump parts separate from those - I use a specific dishpan reserved for that in the other half of the sink, but some people think that's gross, and also then you have to be sure to sometimes wash the dishpan (or mold can grow on it). Put fresh and frozen food away as soon as you use what you need, put cooked leftovers away immediately after eating, and throw wrappings and take out containers into the trash as soon as the contents are used. After a meal, take a wet washcloth and wipe down the counters, and put that in the laundry and grab a fresh one as soon as it seems dirty or within a couple of days, whichever comes first. So have at least several on hand, depending on how often you do laundry. Empty the drain stop into the trash if there's anything in it, and wipe out the sink either with that cloth or a paper towel. (Drains underneath the stopper get gross fast, so that's an instant to-the-laundry for me if I use the cloth.) Unless you're handwashing your normal dishes, all of this is so simple that it should take no more than 5 minutes total on average if you keep on top of it, so I recommend gritting your teeth and powering through it even if the baby is crying for you. But other than putting the food away, it's no big deal to cycle back to take care of this an hour or two or three later. If you formula feed at all, keep especially up on wiping near where you mix the bottles, because mixed formula is vile, sticky, quickly hardening stuff and the powder residue goes everywhere.
You can buy lots of cleaning products for the kitchen, which usage of almost all of them comes down to "spray the dirty thing and then wipe it off with a paper towel or clean cloth," but they're not needed everyday. ALWAYS check the label for the specific materials that product can be sprayed on, because almost all of them have something that it will eventually destroy if used on. Wipes are an easy alternative - basically a combo of same cleaning product and the cloth. Wipe, toss, wash your hands.
NEVER EVER MIX A BLEACH BASED CLEANING PRODUCT WITH AN AMMONIA BASED CLEANING PRODUCT. It literally produces a poison gas, though realistically speaking, I've never seen anyone do it in sufficient quantity to actually hurt themselves. Don't risk this in a place with a baby. Read your labels.
Don't use sponges. They're breeding grounds for bacteria. Green scrubbies, on the other hand, are good, though more meant for dishes than the kitchen proper. Don't use them on anything that scratches, like stainless steel appliances or glass, even though they say non-scratch. Many, many people swear by Mr. Clean Magic Erasers, for both the kitchen and bath, though I've never really gotten into them.
- If there's anything to keep clean in the bathroom, it's the sink (if you're washing the baby there) or tub/shower (if you're using a baby tub or just taking them in with you), and the toilet (general grossness). Once a week is probably the average. The tub and sink use the same sort of "spray and then get off" products as the kitchen (but different ones), but will generally need to be rinsed instead of wiped. Toilets... are more complicated to clean than anything else we've already discussed, and will generally require a toilet brush and toilet bowl cleaner to do inside the bowl, and the other kind of bathroom cleaner for the outside. This seems to be reasonably accurate instructions, though maybe still more complicated than I'd make it (no, I'm not wearing rubber gloves and eye protection to clean mine). If I'm really strapped for time and/or willpower, sometimes I'll just clean inside of the bowl and the seat and under the seat.
- Clean floors are important when you have children on them. On the other hand, your baby won't be going anywhere you don't put them for a long time yet. I hate vacuuming. My vacuum is current sitting in front of the entertainment center, and almost all of my vacuuming is grabbing that from there roughly once a day and taking 30 seconds to just hit the part of the rug where I'm laying my 6 month old to play and the surrounding area (I have cats so I need to make sure the hair is fought back to a certain radius). If your hair is currently collecting into a dustball under your bed, it's not nearly as important as playing with your kid, at least for today, unless you're planning to stick them under there with it. Buy a swiffer sweeper and a wetjet to maintain a minimal level of non-carpet floor and kitchen/bathroom floor cleanliness, respectively, until you're feeling particularly energetic.