To add to what's already been said, yes, Maya can be a bit twitchy about graphics cards. Here's a link with a bit more info on which cards have been certified (you'll have to plug in a bit of info)...
http://usa.autodesk.com/adsk/servlet/syscert?siteID=123112&id=18844534
To that end of the equation, I would simply add that I've had the best luck with ATI (AMD) graphics cards...and not just with Maya. While Nvidia makes some great motherboard chips sets, their "GeForce" vid cards particularly tend to have compatability issues with a fair degree of software...personally I tend to avoid Nvidia graphics cards like the plague (whether they're certified or not).
Also regarding hardware, make sure you system has at least 8 to 16 gigs of ram...Maya can be a real pig when it comes to memory usage.
Now with that I would also ask, when you say "lessons with a lot of plygon counts or animationgs"..ermm...how much is "a lot"? A simple fact of life when it comes to 3D modeling (and especially rendering) is that higher poly counts mean lower system performance. The project I'm currently working on for example has a poly count WAY up in the millions in some scenes...yes, my system is really slow and chunky and some of the renders are taking as much as 10 minutes per frame or more. Even the high end Macs with the Xeon processors (and 32 gigs of RAM) up at the college I attend are struggling with this project. In my case, it's not the hardware, it's simply the project.
While a proper vid card is always a good idea with Maya, my suggestion would be that BEFORE you spend a heap of bread on system upgrades and such, perhaps try running these lessons on another system...see if there's any significant difference in system performance on another machine. If things are still running slow on a different machine, it may be the lessons themselves.
Just something to think about.
Good luck!
