Adore -- you don't need the HD box solely for video games. But, remember that everything is switching to HD in February 2009, so you'll need to go to HD eventually.
If you're looking for a TV that's great for video games, make sure it has 120h. It's a newer technology that cuts down on choppiness when things move quickly on the screen. (This I learned from DH.) We have 2 Samsung TVs and love them.
There are a million myths and misconceptions out there about HD - I sold high-end electronics for 3 years, so let me see if I can dispel some of them.
If you only want to play video games in HD, you don't need an HD box from your cable company. The HD box is only used for cable programming.
The 2009 switch is NOT, I repeat NOT, associated in any way with HD. It is a switch from analog (think antenna) to digital (think cable box). People that do not have a built in receiver or external cable box for their televisions will need an adapter box, but if you're buying a new TV right now, this switch does not affect you in any way, shape, or form.
Depending on how much of a videophile you are and how much money you have to spend, 120hz (which, if I remember correctly, refers to the speed of the video processor in the TV) may or may not be a good option for you. If you want the best picture possible for fast action, then absolutely, 120hz, but it will cost you big money. If you're looking for just an improvement over your regular TV and want to spend a more modest amount of money, just look for the TV with the fastest video processor in your price range. 120hz is not a necessity.
If you already have a cable box and you get a new TV, it's probably worth it to get HD service and switch to the new box. For my HD service, it only is an extra $7/month, which includes the box and a pretty decent number of HD channels. If you want all the HD channels possible, it will be a bit more, but for only a few dollars it's really nice to have.
And also, it will depend on what type of xbox 360 you have as to how you can hook it up to your new tv. Your xbox may or may not have an HDMI input, and in order to have 'true' HD, you'll need one of those, plus an HDMI cable. Some of the earlier 360s only have component inputs, which will still give you better resolution than you currently have, but not quite as good as HDMI.