I would very much like to learn the laws and the practical checking of Shaatnez - how do I go about?
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 Shoshanim offers a 6 month Hilchos Shatnez Course. As their website states:
There are two levels. One is the general knowledge level where one just receives the Shiurim and Level two requires application and acceptance into the program. Once accepted and completed one will have the right to set up a Shatnez Laboratory in their community.