The CPU is the "Brain" of the whole thing so realistically you want to spend around 20% of your budget on this alone. Choosing between Intel and Ryzen is your first decision, after that the decision should be based on what kind of performance you want to get from your PC and what tasks you'll be doing on it (Streaming/ Intense gaming/ Video or Photo editing etc.)

GPUs are incredibly expensive now due to the huge demand and dwindling supply so choosing the right GPU should be based not only on what kind of performance you want out of it (1080p at 120fps+/ 1440p at 60fps/ 4k at 60 fps etc.) Having a GPU is better most of the time than using onboard graphics so opting for a cheaper/ less powerful GPU for now and then selling it later to buy a more powerful GPU later is my suggestion. 

I call the Motherboard the "Nervous System" of the PC. It's equally as important as your choice of CPU but you can find a motherboard for a lot cheaper than a CPU. However, knowing which Motherboard you want should be decided on what kind of functions you want the Motherboard to have (Wifi 6.0/ USB C/ 2.5gb Ethernet etc.) and also if you want the motherboard to allow you to overclock your CPU.

Before you select your RAM you ideally need to know what capacity and speed of RAM would be most compatible for your choice of CPU and Motherboard. Taking the time to properly research this will mean that you won't waste money on really fast, high capacity RAM that your CPU might not be able to take full advantage of. 

My suggestion is always going for a Dual Channel 16GB or 32GB kit if you have the money.

CPU cooling is only necesasry in certain situations mainly if you bought a CPU second hand or if you want to overclock your CPU. CPU Coolers are included with most brand new CPUs and if you choose Ryzen then the cooler is almost definitely going to be up to the task. However if you opt for an Intel CPU 1st Gen - 11th Gen then it's best to put some money towards an Air Cooling solution or Water Cooling solution both having their own pros and cons.

The Power Supply is the lifeblood of your system so it's important to spend a good amount of money on a high output Power Supply with a modular/semi-modular design depending on the requirements of your system. A Gold or Platiumum rated Power Supply is more expensive than standard ones but this will be more reliable than non-Gold or non-Platinum version. Cheaping out on a Power Supply could cause your system issues in the long run or could even short out one of your components if not handled correctly.