To know if you need x86 or x64, you have to know whether the processor you have is 64-bit, which after looking up the specifications for that particular model series, it DOES indeed come with a 64-bit compatible processor.
I forget whether OEM product keys are architecture specific (meaning they will only work as a 32-bit or 64-bit key, not as both) but I know the retail copies ship with both 32-bit and 64-bit architectures, so it may actually work with either. The chances are greater that it was shipped with a 64-bit installation of Windows 7; 32-bit setups are fairly rare with Windows 7 (starter edition excluded). So, download the 64-bit version.
As for working with ISO's, you will need a software package that understands how to work with ISO images, such as ImgBurn or Active ISO(both of which are free - simply do a google search for a download link). Once you've installed an ISO burner, you will need a blank DVD and a DVD-writer/burner.
Generally, software will come with a help file (or online instructions) that can guide you through the process of burning an image. Usually the steps involve opening the software, clicking a button or using the "File" menu to open an ISO image to burn to disc. Once an ISO image is selected (this would be your 64-bit windows 7 ISO file), there is usually a button to "burn image to disc". Once complete, you should have a bootable copy of windows 7 installation media.
As you suspected, you will need your 25-digit product key (found on the Certificate of Authenticity, or COA), and an active internet connection with working driver, or telephone to complete activation. You don't have to complete activation right away, they give a grace period of a couple days, or so.
[WARNING: do not just copy the ISO file as-is to a blank DVD! This will not work or be bootable! You have to extract the files contained inside the ISO file using ISO burning software]
If you get stuck, just report back with whether you were able to download the windows 7 ISO, and what ISO burning software you downloaded/installed, and we can go from there.
