I suggest reading stuff about the game on Boardgamegeek before buying anything. The review are very informative, and most of them detail out what is liked and not liked about each particular game. That being said, you should even try to find a local game store that will let you PLAY the game before you buy it.
And while I'm sure you know this, I will mention it anyway. Just because you like a game doesn't mean everyone else in your family will, so bring them along to playtest with you.
I still have a copy of a boardgame I absolutely love that has never seen a table... Settlers is a great game to get started for a game that doesn't have a roll-and-move component to it.
I myself am into heavy economic games. If you would like to know about heavy economic games, you could ask me! Most people are NOT into heavy economic games.
A couple of lighter games to check out -
Carcassone (or Ark of the Covenant, if you're feeling religious) - You lay tiles to create a landscape of the city, while place people on different features of the landscape to score points, such as roads, cities, fields, etc.
Settlers of Catan (Settlers of Canaan) - Resource gathering/trading game, where you try to build the best settlement.
Ticket to Ride - Set collection game, where you draw cards that correspond to tracks on the board for the purpose of completing certain routes throughout the United States.
You can try Carcassone and Settlers online for free, at link="http://www.brettspielwelt.de"]Brettspielwelt[/link]. Ticket to Ride is also online, but I think it costs money? Not sure.
Sorry, I spend a lot of time with board games, thinking about board games... I want to develop a board game.