I am living in Sweden, but am an American Citizen with an American Passport. Myself and my Swedish family are going to Alanya Turkey for one week on a family vacation. My Swedish family does not require a visa, however do I need one? I do possess Swedish identification.
Yes, as an American Citizen with an American issued Passport, you will need an entry visa to enter the Republic of Turkey. According the Visa Requirements page you should be able to purchase one at your port of entry into Turkey for $30 USD. You will be able to stay for up to 90 days and will not be able to re-enter again for an additional 90 days beyond the initial 90 day limit.
The fact that the rest of your family do not need a visa, are Swedish citizens and that you are currently living in Sweden yourself have no bearing on your need for a visa.
US Citizens do indeed require a visa to visit Turkey, whilst Swedish citizens do not (for a stay of up to 90 days).
Unlike some countries, Turkey visas are little more than a fee payment - at least when purchased on arrival. In purchasing multiple of them I've never had them even look at my passport beyond opening it to a blank page and sticking in the visa.
Historically Turkey has offered "Visa On Arrival" for most nationalities, where you simply paid the fee on arrival at a special booth and were given your visa stamp. However they are in the process of moving towards e-Visas where the visa is requested/issued online. They had previously announced that visas-on-arrival would no longer be available from earlier this year, however they have since reversed that decision at least in some airports, but started charging more for a visa-on-arrival to encourage people to buy them in advance.
Thus your options as a US Citizen are to either apply for an e-Visa on their website which will require a credit card and cost US$20, or apply for a Visa-On-Arrival at the airport which must be paid in cash and will cost US$30.
I would strongly suggest going the eVisa route - not only does it mean a shorter wait at the airport, but it also removes any risk around the possibility they will stop issuing visas-on-arrival before your trip, or that your point of entry will not offer them.