Don't apply for a new credit card. You will likely get rejected and this record will permanently stay in your credit history.
When the cashier at Walmart asks you "Would you like to apply for a credit card and save $20?", the best answer is "No thank you. I'm new in town and I do not have a credit history." Some cashier will not give up and say "You can still try applying for one and see if you get approved. It only takes 15 seconds". You should never say "Ok, maybe I'll try". Instead say "No thank you. Not today. Maybe in a year or two after I build my credit first."
Having said that, you really should 'build' your credit history. Start with a secured card. Most banks will let you park some money (say, $1000 into a savings account) and use that money as a collateral to open a credit card account. They may charge you an annual credit card membership fee ($0 - $150 per year, depending on perks). Once you start using your credit card, you may check back with your bank after a full year, and ask to get the collateral removed, so that you can take out the money for a better use.
Alternatively, you can open up new accounts with no-fee reloadable prepaid credit cards such as Stack and Kobo, where you wire transfer some money first, and then use the card within the limit. These new credit card providers give you nice perks such as $1 off at Starbucks. (Perks change from time to time)
Experian and TransUnion and Equifax are 3 major companies that provide credit history to banks. They also provide paid services if you want to check your credit score, but you can use these FREE alternatives instead.
You can also request to obtain a credit report directly from Equifax and TransUnion once every year. Make sure to uncheck the payment option. You do not need to provide your credit card information to receive this free report.
These services will do a 'soft' credit check (compared to a 'hard' credit check when you apply for a loan or a credit card), which does not negatively impact your credit score.