There is an organization in Lakewood New Jersey that gives a week-long training every once in a while. It is called NCSTAR, WWW.SHATNEZ.N3.NET
I know people from other states that arranged to stay in Lakewood for the week and get trained.
Their contact info is:
You can read a short article about them here.
In addition to @user6591's answer, two additional resources that seem worthwhile exploring:
1) TorahLive.com Laws of Shatnez "Learn What and How to Check - Torah Live" offers an online course along with a certificate of completion.
Not sure what the cost is, how rigorous it is, or whether said certificate is widely acknowledged.
2) Yeshivat Pirchei ...