How do my fellow car-free people deal with moving heavy/bulky items without a car? What are the pros and cons of solutions you've tried?

I'm currently car-free in a city with decent public transportation (by American standards), but things are still very, very car-centric, and also a bit hilly. Living alone, I can manage weekly groceries with a backpack + 1 bag on each shoulder, but it's definitely not my favorite activity. The decent grocery stores are 1 mile (1.6 km) away, so a bit of a hassle to just go to more frequently. For heavy, shelf stable items, I usually try to get those delivered, but it's not always an option.

I also have 2 cats, plus I foster cats/kittens, so I very frequently need to transport animals to/from the vet. I have a backpack style pet carrier, but that doesn't cut it when I have to transport multiple adult cats or a mom + kittens.

I would love to hear other people's experiences, and the pros & cons of various options that you've tried. Some more detailed questions on my mind:

• Do you prefer something you can push or something you can pull?
• How annoying is it to transport when empty?
• How does it hold up to less than ideal weather?
• How does it handle stairs or curbs without a ramp?
• How does it handle poorly maintained sidewalks or unpaved surfaces?
• Is it well made/durable, or something that will probably break in a year?
• If it has pockets or segregated compartments, have those been handy or just annoying?
• Are there any uses that it's not a good fit for?

Edit: any non-bike options? I don't have the space to securely store a bike in my unit, and my building doesn't offer any secure bike storage. Due to all of the hills, I would have to get an electric bike, and was hoping to find some options in the range of $50-$200, maaaaybe $500. For example, I've been looking at collapsible carts/wagons, and pet strollers.