Right, so banks have two factors at play when it comes to interest: rate and number of compounding periods.
For example, lets say you have a bank with 20% interest that compounds yearly (you will not have something like this). You put $1000 into this account. At the end of the year, you get 20% interest, so you end up with $1200.
On the other hand, you have a bank with 20% interest that compounds twice a year (also not going to get this). After half a year, you get half the interest (10%) bringing you up to $1100. Then, at the end of the year, you get 10% again, bringing you to $1210.
APY (annual percentage yield) is just normalizing rate and period into a single number, by setting the period to once and year and adjusting the rate accordingly. Math lets you do this. The first example had an an APY of 20% and the second 21%. APY allowed you to easily compare different rates like 1% compounded daily (about equivalent to 1.005% compounded yearly), 1% compounded monthly (about 1.0045%), and 1% compounded quarterly (about 1.0037%). You'll notice that the period of interest doesn't actually change how much money you are getting by much. Even compounding at an infinite rate doesn't change this (you'd get 1.005%, about the same as daily interest).
If two accounts have the same APY, they give the same amount of money each year, no matter what period each had. Assuming of course, you don't touch the money. If you are withdrawing, there is a very slight advantage to a more frequent period. If you are depositing, there's a slight advantage to a less frequent period.
The money will be applied automatically to your account at every period. You will see it marked as interest in your monthly/quarterly statements or online.
Your calculations show you have 0.06% interest rate, which is reasonable enough for a bank as opposed to a Credit Union (a non profit banking service with limited eligibility and less services that tends to have much better pure rates).
You can find banks with 4% interest, but that should probably make you wary, not excited. In general, the higher the interest rate, the more restrictions and fees that come with, usually minimum deposits amounts and the like.
In the end, unless you are absurdly rich, interest is not something to live off of but rather a manner to sandbag against inflation (you'll still lose value to it, but slightly less). The real advantage to savings is having a savings, interest isn't that big a thing.
Keep interest in mind when shopping for a bank, but also keep in mind other factors, like potential fees, restrictions on withdrawal, and rewards programs, and see what works for you.