Congratulations on your new job!!!
First of all, make sure that they explain eligibility to you. If your eligibility is delayed 90 days or so, which is fairly common, you don't want to miss important enrollment deadlines. All of the "Welfare" benefits (health insurance, disability, life insurance) should be carefully explained to you including all of the 'co-pay' options associated with each plan. If you are young and healthy, you may want to opt to a higher deductible plan. Be sure to take advantage of all disability insurance (short term and long term).. there's a much better chance of becoming disabled than dying before you reach retirement age.
Be sure to ask about retirement benefits and how they are accrued and managed. If you need help deciding whether you can afford to open a tax-deferred retirement account (ex: 403B or 401k), they can help you with the calculations as to how this will affect your take home pay. You will also want to inquire about tuition reimbursement-- how much, when do you become eligible & whether you can use it for non-degree education such as certifications and continuing education.