Tarun's links should get you going. I don't have Illustrator but when I was taking classes the college lab had illustrator and I played around with it. A fellow GIS friend was very adept with Illustrator and blew my mind with how finely you can detail your map�?� you can get it perfect.
Illustrator is very cool for making sexy maps, you have total and complete control over everything. To perfect a map in Illustrator you basically set up your map in ArcGIS to look as close to how you want it as you can, you get your legend as perfect as you can, then you perform a map export to a .svg file (Scalable Vector Graphic), its been a while but I think that was what I used... Now you tweak away until you reach that perfection.
The bummer is that in illustrator you lose all ties to ArcGIS, meaning the data is no longer living data but stagnant. It can't go back into GIS and if the GIS data changes significantly then you are likely creating a new map from scratch in illustrator. The benefit to Illustrator is that you can go into the finest details perfecting your map. A lot of the gorgeous maps you see on display (even many of those that win awards at ESRI conferences) are actually made with Illustrator.
I think the question of Illustrator vs. GIS is really dependent on what your job requires and how much time you can commit to your end product. I work for a municipality and I make many maps that are updated annually or several times a year as the data changes. Exporting to Illustrator doesn't make sense in this situation. Additionally I get flak for spending more than three hours on a map. Everyone assumes "you just print it out from the GIS right?�?� You have to ask yourslef what level of quality does your work environment demand/require?
I'm sure that once you are good with Illustrator and you have your workflows down then the time investment greatly diminishes, however for my work environment I could never justify the cost of the software let alone the time required to master the end product... Boy would I love to though...
Hope this little bit of insight helps�?�