I had a tiny Federal Perkins Loan from my college ($800). It did not accrue interest while in school or during grace period. Same with Subsidized Federal Stafford loans. Unsubsidized Stafford Loans do collect interest while in school and grace.
Just do your research, the rates are not always lower for Stafford loans. I have a few private loans, and two of the lenders have lower rates than my Staffords (though they did collect interest while in school - I made interest-only payments while in school). However, one lender of mine is significantly higher.
On the other hand - if you have financial hardship after graduating college (can't find work, the work you find pays very little, you get sick, etc.), federal loans are easier to defer or put in forbearance to give you some breathing time to find better work or heal. Private loans practically tell you to default on your other loans so you can pay THEM
These people mean business. So weigh the pros and cons with your parents or financial advisor.