tammy,
First of all, congratulations! A few thoughts about your travel plans.....
Six days in the Cinque Terre is not necessarily too long. How much time to spend will depend to some extent on what you plan to do there and the time of year (ie: weather). If you use one of the towns as a "home base", you won't have to "hop from place to place" and that will be a convenient way to both spend some leisure time there as well as explore the local area.
I'd suggest staying in Monterosso as it's the largest of the five towns and has the best beaches, as well as the greatest number of restaurants, shops and other facilities. While there you could.....
- Spend time lounging on the beach.
- Hike the famous Sentiero Azzurro (No. 2 trails) which travel between the five towns (depending on whether they're open at the time).
- Explore each of the towns using the local trains or boats (weather permitting).
- Take day trips to nearby towns such as Santa Margherita Ligure, Portofino, Porto Venere or others.
You could also consider staying in Sorrento and spending time in that area, depending on what type of location you want. The A.C. tends to be a bit more on the "posh" side, while the C.T. tends be more "informal and rustic". Both areas will have lots of tourists. Positano has better beaches than Sorrento, but is a bit harder to get to as it doesn't have a rail line.
The C.T. will probably be the best overall choice, as it has good transportation and you do as much or as little as you feel like. Where to stay may depend to some extent on where you'll be heading after that.