You should be able to get all this stuff new on ebay for $350 or less. Buy a high end (EVGA) power supply used. Don’t buy a cheap new one.
ASRock H110 Pro (you may want to expand)
4MB DDR4 RAM
GPU riser for each GPU
Veddha rack or similar (might as well go with at least 8 GPU size)
Fans for Veddha rack
power supply (size depends on your GPUs and how much you want to expand)
Then you need to choose GPUs. I would lean towards 3 x RX 5700s although that will probably put the cost around $1350.
Once you have the hardware you need a wallet that supports whatever coin(s) you want to mine. If you would rather just get paid in BTC no matter what you are actually mining, NiceHash is a good place to start since they will choose what you mine and all of the mining software is pre-loaded and configured. If you want the actual coins you are mining, then you need to get mining software that supports the coin (Phoenix miner, Claymore, NGMiner, Gminer etc.) After that, you find a suitable mining pool for that coin and point your mining software to that pool. You will configure how often the coins mined are sent on the pool website in most cases. Your wallet address is usually entered into the mining software.
That was the quick version because choosing the best wallet, software, and pools can be pretty research intensive. Mining software and pools have a wide range of fees and different payout schemes so it can be a tough choice.