What 23 y/o doesn't know what a credit score is? How could you not know?
It happens. Chill.
OK, here's the scoop. In this day and age, like it or not, having a good credit score is important.
Not having a score (which is really really rare but it looks like it happened here) or having a bad score will impact:
* Job offers
* Health Insurance rates
* Auto Insurance rates
* Interest rates offered
* Background checks
* Mortgages
* Universal default
* ....
Note that I said "score." I'm talking about the FICO score. You do not learn your credit score through annualcreditreport.com (which is the right place - the place with the damn TV ads is free credit report . com), you have to buy it from FICO (Fair Isaac).
Now, the OP may have what's called a "thin file" or again no file at all, if that's the case you won't be able to get a score from FICO as they won't have any way of verifying data, as there's nothing to verify. This causes a chicken and egg problem, so going to your bank is probably a great move. They know you there and can verify that you are you, and they're probably be happy to set you up with a credit card.
Trying to be debt free? Having a CC is not in conflict with this goal. As others have posted, just pay the bill off in full every month and there is no debt.
The keys to having a good credit score are this:
* No late payments, ever
* Long credit history
* High amount of credit
* Low utilization
The OP won't be able to do anything about the first two except be patient (and not ever, ever miss a payment).
Having a high amount of credit will also take a while. I'm uncomfortable with this part of the scoring as it reeks of a popularity contest amongst those who offer credit, but there you go.
The last part is something that the OP can control once he gets the card. Utilization is the percentage of credit offered to credit used. If the credit line is $1,000 and there are $100 of charges on it, that's 10% utilization, which is nice and low. The FICO scoring mechanism gets "nervous" when it sees people bumping up next to their credit limits.
In sum:
Get a credit card.
Use it sparingly, but use it.
Pay it off every month in full.
There's more to this than that, but the above steps will get the OP going in the right direction.