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To elaborate, I want a wallet that's independent of any third party intervention unlike coinbase wallets being dependent on the company.
If there is the way how much privacy can be maintained?
And is there a way I can access it globally. To be blunt, what kinda wallet would a black-hat hacker would use if at all? Or am I confusing addresses or wallets? Also, it would be better to have a solution in which I won't have to download the entire ledger.
There are no recommendations in this answer, but there's some terminology that will help you with your further research:—
You want a wallet that doesn't require you to surrender your private keys to some third party. Ideally, a widely used wallet for which the source code is published for anyone to audit (i.e. open source), so that there are no hidden security defects.
Such a wallet will eliminate counterparty risk (i.e. the risk that a third party, knowing your private keys, spends all of your bitcoins). However, it will only marginally improve your privacy because all Bitcoin addresses and transactions are visible on the public ledger — Bitcoin is pseudoanonymous. The one slight disadvantage is that you have to make your own secure backups since there's no third party doing this for you. Some wallets let you make a paper backup by writing down a seed phrase — but don't let anyone find the piece of paper, and be sure to have a contingency plan in case of fire.
An SPV wallet allows you to construct transactions without having to download the entire ledger, but an SPV wallet connects to someone else's copy of the ledger, so you would have to decide for yourself if that's an acceptable compromise. After you've constructed a transaction, you can sign it (with your private key) without requiring a copy of the Bitcoin ledger — this is how you spend bitcoins from cold storage.
