ak07, Right now the Army is trying to change its policy on tattoos. If the new policy is approved and takes effect before you enlist, you would not be able to from my understanding. The new policy they are working on prevents soldiers from having visible tattoos while in the summer APFT uniform.
AK07, Here is the current tattoo policy as outlined under USAREC. As noted above though, this policy is set to change. When it will change is not known as of yet. Good Luck!