You said you didn't understand my question, so I'll rephrase. For example, Coinbase Wallet asks for your bank account information as soon as you sign up. Is this the norm? If not, what are better places to use?
Coinbase is actually an exchange, their wallet services are secondary. There are web wallets which do not require such information to sign up.
Or for anyone else that cares to answer my initial question, let me word it differently. I want to remain anonymous at all times.
There is no such thing as complete anonymity in these cases.
That means opening a wallet, funding it, buying bitcoins, receiving bitcoins, and converting to cold hard cash so I can spend the money locally in town. It seems this is not likely.
One does not "open" a wallet; one creates a wallet. 'Buying' Bitcoins is the most problematic part here. You could attempt to find someone dealing with cash e.g. on this website:
https://localbitcoins.com. Similarly, cashing out completely *anonymous* is also a problem. I assume you want to avoid taxes, which is illegal and I shall not give you any advice in this regard.
To create a wallet all you have to do is download and run a client, e.g.:
https://electrum.org/ (no information is required whatsoever).