If you're reading this guide, chances are you probably have a cursory (or better) understanding of the console market and some of the major differences in game titles and general appeal with regards to the three current-gen consoles (Playstation 3, XBox 360, and Wii).  That said, if you're confused, the XBox 360 is basically a Microsoft video game console designed as the successor for the original XBox.  The console has widespread support in North America and some level of support in Europe, but lags by a significant margin in Asian markets. 

Hardware:
You should be aware that a variety of hardware issues CAN plague the XBox 360.  The early issues with overheating and what is now commonly referred to as the "Red Ring of Death" (RROD; e.g. full console freeze/lockdown/breakage) is now remedied by the expanded warranty Microsoft has issued to address this issue. Should you suffer from a circle of red lights within a few months of using the 360, please do take advantage of the warranty (or store return policy).  Do note however, that the warranty via Microsoft will not send you a NEW console.  You will be provided with a refurbished unit, which may or may not have a higher instance of reoccurring problems.  If you're looking for the safest option, take advantage of the Costco return policy/satisfaction guaranteed policy -- the price is somewhat higher than you might find elsewhere, but the safety and comfort afforded by their satisfaction guaranteed return policy alleviates many potential headaches. An alternative solution is to attempt to fix the RROD yourself; that'll generally involve an X-Clamp Fix, but it sometimes works.

Another common issue is what are referred to as DRE's (Disk Read Errors).  These tend to occur when the dvd drive in the console spins up trying to read/access data, and suddenly fails and dies -- spinning down entirely.  This problem will occur on some consoles and varies from game to game.  Many individuals have had issues with DRE's occuring in either: Dead Rising, Oblivion, Gears of War, Rainbow Six, or other similar titles.  The danger of DRE's is not only the annoyance (unpredictable) of the whole process, but also the consequential scratching of dvd discs from the rapid spindown process.  If you should get many many DRE's, please find a way to return the 360 ASAP, unless you enjoy lots of angst.

There are several DVD drives your 360 might come with -- Hitachi, Toshiba/Samsung, or BENQ.  The BENQ is the quietest; if you find that you didn't get the "best" drive (whichever one you want), you might try to swap it for one that better suits your needs/desires.  I'd recommend running a Google search on "XBox 360 Hitachi Samsung" or something akin to that.  Not only will this help you visually ID the drives, you'll get far more info on the specific details that play out the strengths and weaknesses of each drive. If you get a Hitachi, you'll probably want to return it; it's worse in every way.

In terms of chipsets, you might find it useful to try to hunt down one of the 65nm chipsets (Falcon line) -- cooler, quieter, and faster (than the older 360's).  They are slated to be followed sometime later by even smaller 35nm chipsets in later quarters.

Don't bother buying the optional HD DVD drive (unless you really must have it); it's not a requisite piece of hardware for ANY game titles nor 360 related materials and HD DVD is a potentially useless format (in the war against BluRay).  In fact, the only piece of misc. hardware to consider picking up would be an expanded 100GB harddrive (the 20GB HD that comes standard with the XBox 360 Premium is pretty small if you're going to consider using XBox Live Marketplace) and some sort of charging unit for the controller(s).

As to the myriad of editions of the console that are available, you have the option of selecting between the Core (now Arcade), Premium, and Elite.  The Core system retails for $279.99 USD and comes with basic functionality to play games -- everything you need except a hard drive.  In its place you get a memory card.  Happily, it's fairly easy to pick up a discounted hard drive should you desire one at a later date.  The second option is the Premium which basically amounts to the Core unit packaged with the 20GB hard drive; it retails for around $349.99 USD.  The Elite is comes in a matte black finish and is packaged with and HDMI port and a 120GB hard drive; it retails for $449.99 USD. The Core system has been dropped and replaced by the "Arcade" version of the console. Basically, the trade-off will be several arcade games and a larger memory card for the same pricing scheme. You'll also want to check the production line number of the console; 0736 or up indicates the newer Falcon line (65nm chipsets) of XBox 360's -- they run cooler and are less prone to suffering from RROD. The easier way to spot the Falcon system is to look at the box and see the power source; if it's a 175W, then you're golden. If it's not, you're taking a chance.

What Xbox games can I play on my Xbox 360?
You can find a full list of Xbox 360 backwards-compatible games on Microsoft's Web site -- do note that a significant number of popular titles are NOT on the list and therefore don't work; the backwards-compatibility is enabled via software emulation rather than the use of an included chipset for hardware emulation (which is what the PS2 and some variants of the PS3 utilize for broad compatibility). Additionally, Microsoft has started up XBox Originals for downloadable (via XBox Live) titles from the original Xbox console. Note that this only applies to consoles with a hard drive; the Core/Arcade won't be able to play backwards-compatible titles unless you obtain a hard drive.

Games:
It's difficult to keep the game titles themselves up to date in a guide, but it's a bit easier to generalize as to the types of games in comparison to the competition (Wii/PS3).  The 360 mostly caters to the older crowd; there are a limited number of GOOD family oriented games (e.g. I Love Katamari, Viva la Pinata, etc) but for the most part the appeal of the 360 is going to be for more of the traditional gaming crowd.  The 360 isn't the Wii by any means (e.g. broader appeal, party oriented console with traditional gaming appeal) and doesn't pretend to compete in the same circles.  In comparison to the PS3, the 360 (being a Western-based console) has far fewer Japanese/Asian market game productions.  That said, Capcom (Resident Evil 5, Dead Rising, Lost Planet, etc) and Mistwalker (Blue Dragon, Lost Odyssey, Cry On, etc) have been brought on board to fill that niche -- but even then, it's plainly hard to compete against the sheer amount of content pushed out by Sega, Konami, Kojima Productions, Square Enix, Atlus, and the endless myriad of Japanese devs partial to the PS3 and Wii. It has been recently announced that Atlus will be making some games for the XBox 360, so things are looking up in terms of tapping into the Japanese companies. 

Hopefully this short guide provides you with some small measure of help in orienting you towards making a decision as to the purchase of a XBox 360 console.