Many believe that the Servicemembers Civil Relief Act imposes additional requirements for service of civilian process. It does not - there is no federal law requiring that serving military personnel is any different than serving civilians.
Army Regulation 27-40, Chapter 2 governs service of civilian process on military members.
Service off-post is solely governed by state law.
Service on-post depends upon whether the land is exclusive or concurrent jurisdiction:
Generally the provost marshal's office on each installation is tasked with cooperating with service of civilian process, and will be the initial point of contact.
Service of host-nation legal paperwork is typically governed by the Status of Forces Agreement (SOFA) or other agreements between the U.S. and host nation.
Service of U.S. state court paperwork is similar to serving on exclusive federal land - the Army, for example, will first ascertain whether the servicemember will voluntarily accept service. If not, the requesting party is advised that he/she must comply with the requirements of the host nation or Hague Convention on the Service Abroad of Judicial and Extrajudicial Documents in Civil or Commercial Matters. U.S. consular officials will not ordinarily assist in overseas service of legal process. 22 C.F.R. §92.85.
For information on service in a particular country, the U.S. Department of State has a helpful web page with a country-by-country index detailing service (and, if applicable, translation) requirements.
How do you serve a military member who is deployed? Forget about it. Unless the servicemember is willing to accept service of process, or are real close friends with someone in his/her unit who would risk alienating the servicemember by serving him/her (and the command won't help), there is effectively no practical way to serve a deployed servicemember.
Moreover, thanks to the Servicemembers Civil Relief Act, even if you could serve a deployed servicemember, the case would likely be stayed until the deployment was over anyway. So if you're trying to get quick service, as an example to preserve a date for retroactive child support, you better try to serve the person before the deployment.