Hmm... A bit of a tall order, actually. The everyday commands (eg. cp, ps, rm) and utilities (eg. vi, top) are pretty much the same for all Unix-like systems (including Linux) - although the options varies (both available and the letter-switch to use them.
For system-administration on the other hand, they differ widely - from handling printers, adding users and installing packages... even among different Linux-distros, there will be variations here. Further more, the name of devices - especially harddisks, partitions, "slices" - varies. The configuration-files and how services are handled, also differ quite much.
Also the more underlying system - how the disks are set-up (eg. logical volumes or software RAID), how system users and groups are used (eg. Solaris with it's "roles"), and so on - varies much.
As for GUI-programs under X - the "windows" system under Unix/Linux - it will look and feel very differently depending on which WindowManager (provides the shape of windows and buttons and so on) or which DesktopEnviroment (basically a more complex WM with additional programs - editors, mail-client, file-browser and so on - with the same "look", integrated to it) you use - as well as the "theme", as many are themable both for "look" and "feel" (which buttons and how the mouse work). Different Unix-systems or Linux-distros have different default WM/DE (although it's usually simple to switch to another).
Of course when it comes to "daily use", the using of applications - spreadsheets, wordprocessors, image-editing software - the same application will work the same way - especially since even "commercial" Unix-systems now often use open-source software too.
The internals of different Unix-like or Linux systems are similar, but not identical. I wouldn't dive into kernel-internals at once, but if you do, remember they're somewhat similar, but far from identical.
So any Unix-system or Linux-distro should let you learn the everyday commands and let you run (most) applications (runs on the OS, but not part of it - although Linux-distros often bundles both together), and this should pretty much work the same on any other system too. When it comes to system-administration however, although all have basically the same set of tools available, their name and usage will differ between each. Basically, you can learn good admin-routines (eg. taking backups) on any, but the actual commands and config-files must be learned specific for each operating-system.
For a newbie, I would suggest using a Linux-distro - it's similar enough to all Unix-system for the normal stuff, and not more dissimilar for the admin-stuff than any other. It's free and much used, so you will easily find help online. I would suggest you go for one of the three big Ubuntu-distros - Ubuntu, Kubuntu or Xbuntu - as they're much used, have many packages and simple to admin... LinuxMint is also very simple, and is forked from Ubuntu.
When you've learned the basic, you can try the SlacwareLinux, as it's more "hands on" than other distros. There is also GentooLinux, where you build everything from source and "boot-strap" your system while following a book - this process will teach you lots about the system. Finally there is LinuxFromScratch, where you build "your own" Linxu-system (without partial automation like for Gentoo), following a book - this will really teach you how different parts of Linux fits into each other. Compiling a couple of Linux-kernels and looking at the source-code of the kernel and other parts, will also teach you much.
FreeBSD is an open-source free non-Linux alternative, based on one of the "original" Unix-systems (BSD). Solaris is not open-source, but can be downloaded (although updates and support costs) - it's another "original", and has some rather interesting features regarding users (eg. "roles") and disk-handling. Minix is a "teaching Unix", originally companion to a textbook about operatingsystems. Debian/Hurd is a work-in-progress using a very "modern" kernel.
To be allowed to call a OS "Unix", it must be certified... as this cost much money, most Linux and other free OSes (like FreeBSD) are not certified, although most Linux-distros probably could've been. POSIX is another certification-standard, specifying a Unix-like "dream OS". Few (non?) actually follow this standard completely or implements all parts, but many Unix-like OSes are more or less compatible. Again certification costs, so many free project are not certified although they implement the standard equally well as commercial POSIX-certified OSes (eg. the Linux-kernel follows it quite well). Unless you're setting-up a system for the US Government or similar (which may require certification, eg. because of security and encryption concern), I wouldn't worry too much - most of the commands on a Linux-system, is actually follows the POSIX-standard.