You don't "need" a HRM, no, but it is encouraged. I just used mine for the first time and I feel like it gives me a benchmark.
And as for the other stuff Im not certain as it depends what you choose; whether your workouts will be home/outside/gym.
There are a few exercise channels on youtube (bodyrock is one) that you could use.
A lot of the 'gear' is suggested.
The professionally fitted sneakers are so that you are in shoes that fit you properly, reducing the likelihood of injury etc.
You can get them fitted at athlete's foot and them find them on websites for cheaper though. The same as you can use a cheapie HRM and not a polar.

And old member from here did the whole last round without a HRM and just walked/jogged 1hr each day.
The HRM is a tool that may help you learn what your body is doing but i wouldnt say its necessary.