1. A good pair of headphones. But not great...nothing over say $75. Diminishing returns otherwise. A $50 headset should do nicely and be useful for video meetings as well. However, I would not invest less than $30. Your ability to clearly hear what is being said is the most important factor in this job. Normal over-the-ear headphones will be fine as well. I would go with closed back. Ear bud type is not ideal.
2. A keyboard you like typing on. It should be fairly large with good spacing between keys. A membrane type on a small form factor forcing the keys to be small and close together would not be ideal. A larger mechanical keyboard with switches you like would be better.
3. A basic business computer and screen. You don't need fancy graphics. A laptop would be fine. You could even do this work on a tablet using a bluetooth-connected keyboard, but I wouldn't recommend that.
4. Some people use a pedal. The purpose of the pedal is for you to use your foot to stop or continue the recording as it plays. This is so your typing can keep pace with the speaker. Your fingers remain engaged in typing.
5. Assuming you are provided with recording files, I would run them through otter.ai and then clean them up. This would save a lot of time. Otter does cost some money but the annual subscription is only around $100. If you are serious about this as a home-based business that would be a wise investment as it should enable you to earn more money faster.
6. Macros. Not equipment precisely, but recordings of commands linked to a key or combination of keys (eg. F6, CTRL-Y). When you start doing this kind of work I'd expect you to be slow. You will be finding your way along. As you get more experience, you will realize there are certain tasks or commands you keep doing or using. These you can record macros for. May require a higher end mechanical keyboard with its own software, but I'm sure there are open source options out there as well.
The employer may give you access to their own software, where the audio and transcription files will be kept. If you use Otter in this case, you'll get the draft transcript from Otter after submitting the audio file there, download it to your computer, then upload it into their software.
If you are not provided with software, then Google Docs is probably the answer.
You want a physical workspace where you will be left alone and it will be quiet. Noise cancelling closed back headphones will help. You do not want children or animals running through your work area.
Regardless of your starting situation, if this is something you want to do I would begin the work with what you have. If you don't have a great keyboard, or the best headphones, that's all right--as long as they are functional and you can concentrate, you can get started. You can reinvest earnings into better equipment as you go. Don't let a lack of "perfection" stop you.