This is very, very good information.
Disclaimer: I set up my Coinbase account a long time ago, so some of the details may have changed since then.
You can do it all online, but it does need to be attached to your bank account because you have to buy bitcoin with actual money, not with credit. (I think the only option to avoid a bank would be to use cash at a bitcoin ATM, but the ATMs have exorbitant fees and aren't widely available.)
When I set up my Coinbase account, you could either use your bank account (higher spending limit, but longer processing time) or your bank card (lower spending limit, but faster processing time). Your spending limit will go up, and the processing time get faster as you make more purchases.
I also had to submit a photo of the front and back of my ID to verify my identity. They may even require you to send a selfie of yourself. That's fine.
Almost all purchasing methods leave a paper trail, and usually it's a much riskier trail than bitcoin. I can't imagine giving a drug dealer my credit card number or my checking account information! Those directly point to you. Bitcoin doesn't---not directly, anyway.***
Buying bitcoin is perfectly legal. Thousands of people buy it as an investment. More and more legal businesses are accepting it because it prevents customer chargebacks.
So, yes, your bank will know. But they won't care. They probably won't even notice. You're just hopping onto the crypto-fad, right? You and millions of other people.
It's like with receiving shipments. Paranoia sets in. It feels like they must know! But millions of parcels cross the globe every day. We're very, very small needles in a very, very large haystack.
***Ok, this part is risking going too fast and maybe freaking you out a bit, but it is important:
The "blockchain" (the route that your bitcoin travels from you to the vendor) is all publicly viewable online. It's hiding right there in plain sight, because (unlike your credit card or checking account), it's just a bunch of random letters and numbers, not your name.
And nobody but you knows the string of letters and numbers associated with your personal wallet (not your Coinbase wallet, your personal Electrum wallet---move all your purchases from Coinbase straight into your personal wallet first).
When you transfer bitcoin, there will be an option to write "notes." Those notes become part of the public blockchain, so leave them blank. Just skip the notes. You don't need them. Your vendor will give you a unique wallet address for each purchase. That's how they know when you've paid and can link it with what you ordered.
I've actually made that mistake and written the name of illicit drugs in the public blockchain. So, obviously, it's not a fatal mistake. I'm still here to tell the tale. But it royally pissed off my vendor.