A former service member may request a copy of his records from the National Archives and Records Administration (NARA). Usually this consists of the DD214 discharge document (which is critically important), and some administrative papers relating to enlistment, discharge, promotion, schools, awards, and assignments. The longer it's been since your discharge, the more spotty the records. They are much better now than they once were, so current service members can expect better record-keeping than older vets.
Do not expect any medical records from NARA. In compliance with federal law, the VA takes those records. Theoretically there is a method to request medical records from the VA, but in my experience it is highly unlikely that they'll be able to find them for you.
Visit the Veteran's Records site, and follow instructions. Expect something in the mail within 90 days or so. If you don't follow instructions on the page, then you will get nothing.
If you need your DD214 to enlist, go to the site and print out an SF180, and give it to your Recruiter. He may be able to get it quicker through the ARNG Liaison Office at HRC-St Louis, which is co-located with NARA.
Please note that your Recruiter can request a "REDD Report" online that lists your RE and SPD codes, as well as your dates of service and character of discharge, and ASVAB scores (newer Soldiers may also have PULHES codes from enlistment physical). This document will legally substantiate ASVAB scores only, and thus will not substitute for a DD214 or military physical.
If you have problems or questions with this procedure, don't contact me -- I'm just the messenger. Best wishes!