Vacuums under $100 are generally very low quality with poor capture and filtration abilities. There are several good vacuums in the $100 - $200 range that are worth mentioning, such as the Shark Navigator and the Oreck Quest.
Bissell and Hoover both make relatively inexpensive carpet cleaners that work well. Our tests of the Hoover SteamVac show it to have excellent cleaning abilities on light to medium soiled carpets
For heavily soiled carpets I would recommend the Rug Doctor Mighty Pro, or professional cleaning.