After upgrading to Snow Leopard, don't I have to upgrade Chrome or Firefox and other programs to 64-bit version?
64-bit apps have been fully supported since Leopard (OS X v10.5), and 64-bit non-GUI programs since 10.3.  The only thing 10.6 added was that Apple included a 64-bit kernel (which is a completely separate thing), and included 64-bit code in (almost) all of the programs they ship with the OS.

OS X uses a "fat" binary format, which allows a single program to include multiple CPU-architecture versions of itself; when it's run, the OS picks the "best" version of the program, and runs that.  For instance, on a 64-bit capable Intel Mac, it'd look for a 64-bit Intel version first; if that wasn't available, it'd look for a 32-bit Intel version; if that wasn't available, it'd look for a 32-bit PowerPC version and run that under the Rosetta emulation/translation system.

Net result: many of your programs are probably already 64-bit capable, and have been running in that mode for a while now.  Some may not be, in which case they'll keep running just fine in 32-bit mode.  It's all remarkably transparent.

If you want to know which of your programs support 64-bit mode, run the System Profiler utility, and select Software -> Applications from the sidebar; it'll list all the programs it can find, along with whether they're Intel-only, PowerPC only, or Universal (both), and whether they have Intel 64-bit support.  You can also use Finder's Get Info on a program; if it has an "Open on 32-bit mode" option, that program includes both 32- and 64-bit versions of itself (and that option will let you override the OS's ideal of the "best" version).

Now, to answer your more direct question: I don't know about Chrome, but Firefox version 3.x comes in a 32-bit only binary; version 4 comes with both 32- and 64-bit.  Either one will work fine.