Both of you would benifit greatly from a reproductive short course... many state universities offer them including CSU (where I work...
so I there is some self-interest in there as well).
As far as the actual AI on the mare's side, its very easy with the proper restraint (you want to insert your arm into an uninterested mare?). Wash her with Ivory, put on a sterile glove with a little sterile lube, insert your hand, guide a sterile pipette through her cervix, deposit semen, come out, all done
As far as the collection, that requires a little more effort, training, and equipment. Worst case you can bring your stallion to a breeding facility and they can handle it for you. There is a lot of very pricy equipment needed, starting with a good microscope, warmplate/incubator, and hemocytometer. Together that's $500-2000. Add in all the mechanized fancy equipment and that price shoots up to $5000. That equipment is needed to keep the semen and everything that touches the semen warm, count the concentration, and look at the motility. If you only do one or two artificial breedings, its far simpler to go to a local breeding facility to handle it for you. Just pass the fees on to the mare owner; they should expect it.
As far as fertility testing, there is only one way to do that: breed mares. We can ESTIMATE fertility, but there is no other way to test it other than test breeding. As far as evaluating semen, I will try to give you a basic overview of what we do for every stallion we collect.
1) Bring the av into the wash room, take off the protective cover, and remove the gel fraction from the ejaculate. Pour the semen into a warmed graduated cylinder and measure the gel-free volume. Any live sample of semen needs to be kept at body temperature until its extended with a semen extender (they don't just ship semen, it MUST be "extended" in a medium that protects the cells, feeds them, buffers waste, and protects from temperature changes.)
2) Measure concentration of the sperm. You can use a hemacytometer to physically count it on a microscope (hemacytometer = $100 or so, reusable, takes upwards of 10 minutes of work) or use a densimeter to use light to measure the concentration ($1500-2500, takes about 30 seconds).
3) Dilute the sample to about 50 million cells per ml, and estimate total and progressive motility on a microscope.
There are also other things we can do in a research setting for stallions that present with symptoms of infertility.
4) Flow cytometer: uses florescent stains and a laser to measure a variety of parameters, including live/dead (are they dead, or just not moving?), mitochondira (is their energy souce any good?), membrane fluidity, etc.
5) Morphology: we do this for every breeding soundness exam; stains the sperm cells and then look under a high power microscope to examine the cells for defects like broken tails, cytoplasmic droplets, etc. There are also even better microscopes that can see pitting on the sperm heads, and even more powerful electron microscopes that can examine DNA, organelles, etc.
There are a lot of other things as well, I just don't do them often enough to know
Yes, you can send a sample to a lab for some of those things, morphology especially. If you aren't very close then some of the other things aren't really possible, but most are
If you can get your stallion to a reproduction clinic where they can ship the semen to a lab that'd work too. What exactly are you looking to measure/test/evaluate?
