I believe his model is:
Gateway E-6610D and it contains
667 Mhz DDR2 memory.
*** crucial memory specs ***
* This review is for the E6610D SB. The SB comes with a 10,000 RPM drive and possibly other perks. I don't think our user has this edition.
http://www.crucial.com/usa/en/compatible-upgrade-for/Gateway/e-6610d
*** PC magazine review ***
http://www.pcmag.com/article2/0,2817,2069888,00.asp
ebay query: 8gb ddr2 pc2-6400 desktop
http://www.ebay.com/sch/i.html?_odkw=2gb+ddr2+pc2-6400+desktop&_osacat=0&_from=R40&_trksid=p2045573.m570.l1313.TR0.TRC0.H1.X8gb+ddr2+pc2-6400+desktop.TRS0&_nkw=8gb+ddr2+pc2-6400+desktop&_sacat=0
* According to Crucial, the computer holds a maximum of 8 GB (2 x 4GB).
* I "think" the computer is only SATA 1 (1.5 Gbps); one-fourth the speed of modern SATA
* I "think" the computer is only PCIe x 16; 1st generation. Meaning that you can NOT use a PCIe 3.0 card.
* I "think" PCIe 2.0 is compatible with PCIe (v1.0).
*** NewEgg PCI Express 2.0 x16 cards ***
http://www.ebay.com/sch/i.html?_odkw=2gb+ddr2+pc2-6400+desktop&_osacat=0&_from=R40&_trksid=p2045573.m570.l1313.TR0.TRC0.H1.X8gb+ddr2+pc2-6400+desktop.TRS0&_nkw=8gb+ddr2+pc2-6400+desktop&_sacat=0
*** Recommendation ***
Were I in your situation, I wouldn't consider upgrading an 8.5 year old computer. But since you stated that this is your first computer, I'm assuming that you're young in age, or in a part of the world that new computers are hard or impossible to come by. If you're insistent that you want to upgrade this computer then I would suggest buying 8 GB (2 x 4GB) from Ebay. Chances are that a lot of that memory is selling out of Hong Kong, which means that there could be a 2-4 week time from order till delivery. Also, you could purchase one of the cards from NewEgg or purchased a used card from Ebay as well. Normally I recommend that most people upgrade from hard drives to SSD's, but since your new/used computer only has SATA, you might only see a seq read speed of 150 Mbps. My mom's computer had a hard drive crash on a ten year old computer that she's committed to, and I added a Samsung 850 EVO 120 GB. It's an upgrade, but nowhere near the performance that the drive afford. I don't even think her computer has AHCI. Anyway, let us know if we can be anymore help.