I am 24, and my tests indicate I'm at high risk having a Down syndrome baby. Should I have the amniocentesis?

Depends. Anyone is at risk for a baby with down syndrome. If testing shows that you are at high risk, the risks of having a baby with down synedrome are higher than the risk of pregnancy loss. The loss with amniocentesis is only 1 in 1600 in experienced hands. The only way to know whether your baby is affected or not is to have an amniocentesis.
Other methods too. They can do a quite specific test on fetal cells that circulate in your blood.Initially developed for paternity testing, the NIPP can be done as early as 8 wks and can also be used to study for Down syndrome or genetic issues.
Depends. Here is the delicate problem of this tests: if you really dont want to have a baby with down sdr and you are willing to interrupt the pregnancy, then go ahead and get the test done. If you will maintain the pregnancy no matter what , getting a positive test will make you live with guild and negative feelings for the rest of pregnancy. Best thing is to get a maternal fetal medicine specialist opinion.