It depends on your financial situation, but $300 is a solid starting point if you can manage it.
When you first arrive, the facility issues necessities, but basic is the right word for it. The soap, shampoo, and toiletries they hand you are the bare minimum, and most inmates replace them with name-brand versions from the commissary as soon as they can. It is a small thing that makes a real difference in daily comfort.
Beyond toiletries, there are a few items that matter more than people expect going in. A decent pair of sneakers is at the top of the list. You are on your feet constantly, the issued footwear is often of poor quality, and having something that actually fits and supports your feet is worth every dollar. A radio is the other essential. It is your primary entertainment, your connection to the outside world through news and music, and in a place with a lot of dead time it becomes genuinely important to your mental state.
That $300 breaks down roughly into commissary essentials, a phone account with enough to make initial calls home, and whatever the facility allows in terms of personal property purchases. If money is tight, prioritize the phone account first, so family knows you arrived safely, then work toward the other items as funds come in.
Having money on arrival also signals to the people around you that you are not completely without resources, which matters more than it should in that environment. Going in with nothing makes the first few weeks harder than they need to be.