First, if you live in/around a reasonably populated urban area, and you're in the United States, I can't see why you would choose to bank with Chase, B of A, or another large commercial bank. I think you would be much better served by banking at a reasonably large credit union.
There are many differences between banks and credit unions, but in a nutshell, credit unions are owned by the members, and operate primarily to provide benefits to their members, whereas a bank is owned by the shareholders, and operates primarily to make profits for the shareholders (not to benefit the customers). The banking industry absolutely hates the credit unions, so if you've ever been nickeled-and-dimed with this fee and that charge by your bank, I have to ask why you're still banking with a company that irritates you and/or actively tries to screw you out of your money?
I live in California, and I've banked at credit unions almost exclusively since I started working nearly 30 years ago. Every time I've strayed and started banking at a for-profit bank, I've regretted it.
For example, a few years ago I opened a checking account at a now-defunct bank (WaMu) just for online use: eBay and so forth. It was a free checking account. When Chase bought WaMu, the account became a Chase account, and it seemed that every other statement brought new fees, new restrictions, and so forth. I finally closed it when they imposed some stupid fee for not carrying enough of a balance. I found out by logging in to their Web site and seeing a balance of zero dollars; they had imposed the fee a few statements back, and I had missed it, so they kept debiting my account until it was empty.
At this point, I do about 90% of my banking at a fairly large credit union. I have a mortgage with a big bank, but that was out of my hands, as the lender/originator sold the mortgage and I had no say in the matter. My credit union has a highly functional Web site, permits me to download my account activity to Quicken, and even has mobile apps which allow me to deposit a check by taking a picture of it, or check my account activity, etc.
They (my credit union) are part of a network of other credit unions, so as long as I am using a network ATM, I never pay a fee.
In sum, I can't see any reason to go with a bank.
Regarding checks, I write a small number of checks per year, but I recently needed to reorder them. My credit union refers members directly to Harland-Clarke, a major-league player in the check printing business. Four boxes of security checks was around $130 plus shipping, which is not small money. However, I was able to order the very same checks via Costco for less than half that amount. Costco refers members to a check printing service, which is a front/subsidiary of Harland-Clarke, and using a promo code, plus the discount given for my Costco membership, I got four boxes of security checks shipped to me for less than $54.
My advice would be to look around. If you're a Costco member, use their check printing service. Wal*mart offers a similar service to anyone, as does Sam's Club, and you can search around to find other similar services.
Bottom line, if you order your checks via your bank or credit union, chances are you will pay full retail. Shop around, and save a bit.
I've not opened a new account at a credit union in some time, but I would not be surprised if a credit union offered a free box of checks when you open a new account with them.