Your banks does not have to be listed on Coinbase to use it. You add your bank account number and routing number, they make 2 small deposits, then you confirm those deposits to prove it's your account and they are linked.
Your bank card may or may not work. I've had multiple banks cards added to my account, from the same bank even, and some worked, some didn't. It can be annoying to set up, but once you get them, you are good.
You will have limits regardless. It's a must to prevent fraud, because once you purchase bitcoin instantly with your debit card, you can send the bitcoin off coinbase, and if it was a fraudulent transaction, they'd lose too much money.
The ONLY WAY TO UP YOUR LIMITS is to keep purchasing bitcoin. The more you purchase, the more it proves it's you making the purchase with no fraud or chargebacks. They WILL RAISE YOUR LIMITS once you have a couple of purchases.
Coinbase can be tricky and annoying but I've been using them with very little issue since 2012. The main issue I ran into was if my bank card was canceled due to expiration or fraud, and the banks replaces with a new card, sometimes you don't see the 2 small deposits to verify. Contact CB support and contact your bank. They will both work with you to resolve.
Once you have purchased Bitcoin or Ethereum on CB, you are free to send it to an exchange to trade for Steem or SBD. It will be great when CB adds Steem and SBD support one day.
One last tip: use coinbase.com to sign up. It's much less frustrating to do on the website than it is on their mobile app :)
Good luck!
Awesome, thanks for the help! I will definitely look into it.