Which Gentoo should I download?
You'll probably find the "Choosing the Right Installation Medium" part of the install guide helpful.
install-x86-minimal-timestamp.iso is the LiveCD image. You burn that to a CD and boot off it to get into a minimal Gentoo environment that you can use to install Gentoo on your actual system
stage3-i486-timestamp.tar.bz2 (i686 for pentium 2 or greater) is a tarball of the root filesystem; it has /dev, /bin, etc. One of the first things you'll do after booting the LiveCD and partitioning your disks is to download the stage3 tarball and unpack it
The file is called stage3 because you can technically start installing Gentoo from any of three stages, depending on how much of the build you want to do yourself (e.g., stage2 doesn't include a base system -- you need to compile it yourself). stage3 is the only one officially supported, and almost certainly what you want to use
The "minimal" is the CD image that just acts as a live distro for you to perform the Gentoo installation on -- it holds no (or very few) packages, so an Internet connection is still required while on that CD. Today, many Gentoo installers use System Rescue CD instead of the Gentoo-provided images. This step is really only for those that want to install Gentoo onto their current bootable partition.
stage3 is a "higher level" of the old stage 1s. Stage 1 was the "base" that you would start compiling from. Today, the stage 3 provides more already-compiled packages, so the system is a lot easier to set up. Stage1s would require users to do a lot of manual bootstrapping (recompiling with the application you just recompiled... confusing, right?).
You will need to grab the stageX once you are in the environment that you want to start building Gentoo, and it is highly recommended today to start with the stage3s, as opposed to stage 1s. You extract it, and then start building on top of that. See the Gentoo Handbook for instructions on building from the stage3 tarball.
Some like to "recompile" the stage3 binaries with their new USE and CFLAGS, and that is done by bootstrapping.
Good luck installing Gentoo! That's what really got me into GNU/Linux, and I hope it does the same for you.