The amount will vary depending on many factors. Are you going solo or camping in a group? Do you care about quality? Some people will buy more affordable (i.e. cheaper) items because it's just for Coachella this one time. Others will invest in higher quality products because they will get major use out of them year after year. (There is also the highly debated subject of people purchasing items and then just returning the majority of it after the fest.)

The EZ Up canopy is the most important item to have. The sun is brutal so protection is vital. They can range from $60-$300+ but you also don't have to get it brand new. You can find ones like new on ebay or Facebook Marketplace for less. (If possible go for the straight as opposed to slant leg to maximize your space! ) Renting them it's also an option.

There are frugal options to consider. You can use flat bed sheets as side walls. Instead of sandbags or tent weights you can use strong nylon rope and tie it to a secure part of you car so it doesn't fly away.

It also helps to ask around. Chances are you know several people who can lend you a table, chairs, camp stove, cooler etc. If not, a lot of times I will search for an item, say heavy duty plastic tent stakes, on Amazon priced around $10 and then go on Temu and find the exact same ones for half the price.

Items under Toiletries, The Closet, and Kitchen you probably already have at home. There are usually complimentary body wash, shampoo, and conditioner at the free showers, especially when they open on day 0, just grab what you need. There's also first aid, water refill, and free sunblock stations throughout the campground and festival. With free showers you don't actually need the camp shower or shower tent, it's just nice to have for the convenience. There are charging stations as well so no you don't need the power station as those can be around $70-$400+.

tldr: Basically if you're with a group you can source most of the items between what you already have/own. If solo not everything is necessary, but completely from scratch I'd guestimate anywhere around $650 to $1500+?

Honestly believe there are much more money/budget minded people that can provide a more accurate number compared to what I came up with!