1) Get added as an authorized user to your parent's cards (make sure they ask for your SSN). These will report on your credit score and increase your amount of available credit outstanding, thus lowering your overall credit utilization (30% of score). This only makes sense if your parents have never been late on the card and keep utilization low (under 10%).
2) Check your credit report with the 3 credit bureaus for free at annualcreditreport.com-- you are allowed to access your report once every 12 months. You will not see your FICO score, but it is always valuable to check for mistakes or errors on your report.
3) The amount of credit you get approved for initially is mainly a function of FICO score + stated income. Notice stated income. The credit card companies do not verify your income. Sometimes they will ask to access your tax returns (very rare) and you do not have to provide it to them (app will be declined). Every time you request new credit--they will Hard pull you (negatively effects your score). Sometimes they ask for Household Income--when I was in college I just included my parent's income. Never had a problem. The goal is to get the highest available amount of outstanding credit allowed. This will reduce your overall utilization rate, thus improving your score dramatically.
4) Chase is notorious for Hard Pulls every time you ask for a credit line increase which is why I don't like them. American express however will SP credit line increases. Blue cash everyday 3% on supermarkets, 2% gas, 1% on everything else. Furthermore, AMEX has a special 61 day 3X credit limit hack available. Once you open your new account (HP)--you are eligible for a 3X increase to your credit limit after 61 days--best of all Soft Pull (does not impact credit score) guaranteed. This makes it much easier to increase the amount of credit available--which helps increase your score. Just don't go over 25K or you risk a Financial Review (they temporarily suspend you card and make your prove income, assets, etc).
Another benefit of AMEX is backdating your card. I got my first AMEX 8 yrs ago. Got another one last year. They back date your new card so that it looks like it too has been open 8 yrs. This increases my average age of credit (about 15% of FICO score).
5) Don't forget Discover. They have a new product with their Discover IT card which allows you free access to your real FICO score once a month. They have rotating 5% cash back categories which change quarterly, along with 1% back on everything. Also they will SP credit line increase requests after 3 months. I have this card purely for the free FICO. Some companies scam you buy charging you for your FICO score--this way I get it for free. This card definitely pays for itself.
6) Hit up the MyFico forums--they are the best when it comes to all things credit cards and improving your FICO score.