What kind of SD card will I need (to save Linux onto) and where can I buy an ARM processor (one that does not come pre-soldered onto the board like Raspberry Pi)?
Head over to Olimex, and they have a line of products called Olinuxino - with options based on ARM9 SoC (a Freescale i.MX 233 one running at 454MHz), ARM Cortex-A8 SoC (Allwinner A13) and I believe there are a few in the works based on even higher end SoC's (Cortex A9/A15).
One of the design guidelines of Olinuxino (as I understand) is that their board designs are hobbyist friendly, and for a better-than-newbie hobbyist, ability to do the PCBA using hand-soldering is part of such goals. I believe, all of their boards, are open-source HW, so that may be a good starting point.
Freescale ARM9's might be available on Mouser (or the other online distributors), but Olimex might be able to provide some hard-to-find parts, for example, they do sell the Allwinner A13 and it's matched power-management IC.
Anyhow, assembling a linux capable board requires access to tools, techniques, and skill, only fairly advanced hobbyists might have, and being able to design such a thing from scratch, IMHO, is not quite in the hobbyist territory.
Yet another alternative could be something like ARM7TDMI SoC's, although they can only run ucLinux, I believe. I'd read somewhere that Linux kernel trunk had support for MMU-less processors added, which means, that a tiny distribution of standard Linux built for ARM7TDMI might also be feasible. Typically, ARM7TDMI's processors would be available in far more hobbyist friendly packages, with far fewer pins, and reference designs around them -- I'd expect to be far simpler. If you search for the like of Blueboard (IIRC!), you might find ready to use board, but also it's schematic. PS> I got the ARM7TDMI thought, thanks to your ARM7 tag.
While as noted in comments to my original post, you explicitly did not ask for a pre-integrated board like the Raspberry Pi, Raspberry Pi is the way to go if you want to start working with ARM. Jumping straight to an unintegrated chip is probably a lot more work than its worth.
Freescale IMX.6 if you are looking for something with faster graphics.
Both will require an SD card. Grab one that comes as microSD with a SD adapter for maximum compatibility. Recommend going with at least 8GB
Edited to fix reference to miniSD, which was incorrect.