I encourage you to keep a checking a savings account for your primary banking needs. Choosing the right bank or credit union depends on the interest you can earn and what free services are offered. Today, most basic checking accounts provide most services free of charge, but the savings accounts typically offer very low interest rates. As such, keep only the minimum amount of cash on hand at this institution to ensure you are not charged any fees as well as to cover your cash needs for at least a couple of months. The cash need can be determined based on your monthly expenses and should be there in the case that your income is stopped (due to loss of job or other emergency).
The remainder of your money should be invested wisely. Enough should be kept liquid in money markets to cover for a rainy day or rainy 6 months, but this can be at a larger institution such as Vanguard. Transferring money to your bank accounts can usually take place in a couple of days so the cash is close at hand, but is earning a much better rate of return. Vanguard is one of my favorite institutions as they have a wide range of investment choices and possibly the lowest fees of anyone.
Longer term investments are a whole different can of worms so I'll open that one up later in the future.