Getting started as a first time home owner

I’m a first-time homeowner with a small lawn out back, and I’m looking for advice on how to get started with lawn care. The space isn’t huge (townhouse-sized), but right now it’s pretty overrun with weeds, and I’d love to tidy it up and get it looking nice. Photos of the lawn here: https://imgur.com/a/wMQGaAw

Since I’m totally new to this, I have a few questions:

1. Weed Removal: What’s the best way to tackle the weeds? Should I use a spray, pull them by hand, or try something else?
2. Lawn Revival: Once the weeds are gone, how do I encourage healthy grass growth? Do I need to overseed, fertilize, or do something else? I try to mow every second week but sometimes more often in Summer.
3. Tools: What are the must-have tools for someone with a small lawn? I want to avoid over-buying but also don’t want to be caught unprepared.
4. General Maintenance: Any tips for keeping it tidy and preventing the weeds from coming back?

I’m in south-east Queensland so it's the middle of summer and the lawn is growing quite fast! Any suggestions that take the climate into account would be awesome.

Thanks so much in advance! I’m really excited to make the space look better and learn more about lawn care.