You said that you have an "offer" for a credit card...is that a pre-approval or a prequalification? There is a difference.
Prequalification simply means you meet the bank's general underwriting requirements to qualify for an account, but that doesn't mean you'll actually get the card. You'll still go through the evaluation and approval process, and actual approval rates can be low because the bank is simply targeting a demographic that meets a set of guidelines.
Preapproval means the bank is extremely likely to open an account for you, but there's still a chance (small, however) that you'll be denied.
Either way, you will probably see a hard inquiry on your credit report, which can have a short-term negative effect on your credit score. The amount of that effect has quite a bit to do with what your credit file looks like now, and from what you've shared, you don't have anything there now, so inquiries will have a larger effect on you than they would on people with more established credit histories.
The point is, don't take on inquiries you don't need to, and try not to have very many around the same time. It can make it look like you're shopping for credit, and that can be a red flag for some lenders.
Another final point here - credit will be a bit of a challenge for you right now because you don't have any established credit AND you are just starting your first job. It would be helpful if your parents were willing to co-sign a car loan for you so that you don't get hit with a high interest rate. Most lenders who might approve you right now based on your circumstance will either require a high down-payment or a high interest rate because you would fall into a high-risk category. If your parents are willing to co-sign then you'd have a better chance at a smaller down-payment and less of an interest rate (assuming, of course, that their credit is good).
Take your time, research your options, and then make a move. If your parents are willing to let you drive one of their cars for a little while then maybe that would be a better option while you settle into your new job, save some money, and decide on how to get started in credit. Keep this in mind - if you're going to shop for car loans, do it in a short period of time. Credit laws require the bureaus to count multiple inquiries for mortgages and car loans in a 30-day window as one inquiry, so you don't have the negative consequence of each individual inquiry hurting your credit score. This lets you shop for the best deals before you decide.