I don't like asking coin base to create an account for me and having them give me my private key. How can I do all of this process myself? Either bitcoin or ethereum
If you want to do everything yourself, download a wallet program on your computer and run it instead of using coinbase. For example, if you use a wallet like Bitcoin Core, it is completely under your control on your own computer, and its completely open source so you can review it all yourself. Thus you don't have to trust any third parties like coinbase. That is the principle behind why bitcoin is designed the way it is, and bitcoin core is the evolved version of the original bitcoin software Satoshi wrote to be trustless in this way.
having them give me my private key.
As Pieter mentioned, coinbase will not give you any private key.
for now just open an account. i don't know if i could not use a wallet and manage my money myself.
Just to clarify a nit with this point, bitcoin doesn't have any "accounts", in the way you might be thinking. You just have private keys, which correspond to addresses, which bitcoins can be paid to. There is no "balance", just unspent bitcoins sent to addresses, which are known as UTXOs (unspent transaction outputs). By installing a wallet on your computer, it will store your private keys for you, and watch the blockchain for transactions sent to your addresses. It doesn't ever "register an account" anywhere :)
btw.is there a way to purchase btc and make sure that i will get what i pay for? i guess i should trust coinbase for that one. but maybe you know of a better way
Currently you just have to trust a seller like coinbase, yeah. If you pay via credit card and don't recieve it I imagine you could contact your credit card provider and ask them to reverse the payment or something though, not sure. But yeah there's no way I know of to trustlessly exchange BTC for fiat currency.