Congrats on your new home!

I would suggest an expandable pole duster to sweep the dust off the ceiling/corners, ceiling fan blades, etc. These are sold everywhere and I don't have a specific product recommendation. Just remember to take proper care of the microfiber attachments (see below)

A shop vac from Lowes/Home Depot/hardware store. Not something you'll use in your regular cleaning routine, but invaluable to have for emergencies and specialized tasks.

Speaking of emergencies, this plunger is amazing. The picture showing all the attachments makes it seem complicated to use but it's not, you won't need to use most of those attachments.

I'm not sure how much (if any) carpet is in your house but if there is carpet/rugs, I like to use scented carpet powder like Arm & Hammer or Glade.

A stiff broom or two for outside/garage use only. A softer more pliable broom for indoors.

I find the Swiffer really comfortable and easy to use especially in larger spaces. You don't have to keep buying pads for it, use microfiber towels instead. They can be reused over and over.

Microfiber cloths in general are very useful for cleaning jobs all around the house, you can get a big pack of them on amazon, look for the one that has like 80,000 reviews. I color code mine: blue is always for bathroom, yellow is always for kitchen, other colors are for whatever other tasks; i personally don't care to color code further than blue = bath yellow = kitchen.

Follow the care instructions for your microfiber towels or they won't work nearly as well: Don't launder them with any items but other microfiber towels (because they will attract the lint from clothes and bath towels if you have them in the same wash).

Don't ever use scent beads or fabric softeners when washing microfiber towels, these products contain polymers that coat the microfiber, making it far less effective. I have a bottle of completely unscented laundry detergent for my microfiber towels and I know the bottle will last forever because I use so little product laundering my microfiber.

Don't wash microfiber in hot water (fibers can melt/deform). Avoid machine drying but if you insist, keep the temp LOW. Best to air dry them.

You can go several uses between laundering microfiber towels, just rinse with water and squeeze dry after each use.

If you keep a microfiber towel next to your shower and wipe down the walls/tub after every shower it cuts down immensely on mildew/soap buildup/shower grime.

Properly maintained microfiber works great with just water. The only all-purpose cleanser I keep on hand is 409. If I feel the job needs something stronger than just water + microfiber, I will use 409 or Dawn foaming dish soap spray.

If it's something that needs to be disinfected, I dilute Pine Sol (ammonia) with hot water and add some Dawn to it. This works similar to 409 so you probably don't even need both, but the latter is cheaper because for properly disinfecting you need to really saturate that surface and it's cheaper to do that with diluted ammonia vs 409.

As for specific-purpose products, I use Easy Off oven cleaner because it frigging works and works fast (don't care that it's cHeMiCaLs). I keep Windex on hand because I like it for the mirrors, windows, and kitchen appliances. I keep Mr Clean erasers on hand to get scuff marks off the walls, trim, baseboards, doors, etc. I'm very gentle with the erasers because they are abrasive. Also Barkeeper's friend is a gentle abrasive in powder form that you can sprinkle over the counters, sink, etc.

I don't use bleach as part of my regular cleaning routine. I do have some products that contain bleach, such as gel toilet bowl cleanser and clorox wet wipes. I use those for the toilet. Otherwise I prefer ammonia for disinfecting because it's better at cutting grime, it doesn't discolor fabrics, and the scent is much better than bleach (I like the lavender Pine Sol).