One of the recent changes with Cricut is that they went from having a web-based design space to an installed app (Mac compatible). I've only heard good things so far about this change and, IMO, is the final push for me to decide to go with it as my cutter and pen plotter.
I've researched and compared to several brands (brother, Siser, Silhouette) and have used the Cricut at work for cutting vinyl. The vinyl cuts are superb and so far has never had an issue.
I imagine the most rigid thing you could cut with a cricut is Balsa. I work with wood every day and I'd say balsa sheet is a fairly good construction material for models and display pieces.
Otherwise you could look at vinyl and polyester sheets for plastics.