Welcome to the forum. The first thing I would do is request a copy of your credit report and FICO score to see where you are in terms of your credit. This should really be done annually to ensure that there is nothing incorrect on your bureau.
However, if you do not want to go this route, AMEX has a pre-approval function on their website: https://www304.americanexpress.com/secure/my-special-offers?btct=SJHP_hero_ultimateapply
You will need to enter the information requested. This will result in a soft-inquiry which means that nobody will see the inquiry on your Experian other than you. After the results are in, AMEX may or may not offer you various cards that you may qualify for. Again, this is a pre-approval function and is not a gaurantee for credit.
Because you are newer to credit you may want to look into the AMEX Zync card. It does have a $25 annual fee; however, it gets your foot in the door with Amex. After a year or two of building your own credit, you may apply for an AMEX credit card such as Blue Cash etc. Because you became a Member in 2011 (hypothetically if you apply and are approved for a Zync this year) your new AMEX card will be backdated. What this means is that your new account will report as opened in 2011 even if you opened it in 2013. AMEX is the only company that does this.
Do you have any student loans? A car loan? Or is your father's Amex the only item on your credit report?
As much as I admire your stance on only having one credit card, I would advise having a Visa or MasterCard as back-up in the event you cannot use your AMEX. A debit card could suffice if you have enough funds in checking to cover anything in an emergency.