There are lots of ways you can teach yourself to identify what might be ailing your animals......without needing to be qualified

I have a sort of mental flow chart for my pigs, it goes a bit like this...
Is there an obvious problem or do they just seem a bit 'off'?(If there's an obvious problem like lameness, injury etc then I take the appropriate steps - cuts/wounds get cleaned, sprayed with antibiotic spray and if possible a dressing applied. Lameness involves an examination to see if I can identify what/where the cause lies....depending on what I find it's usually a phonecall to my vet to explain and then a run over to collect any medication - anti-inflammatory etc.)
Do they have a low/high temperature?Are they eating/drinking/pooing normally?
Is there anything in their environment which could cause problems?(Have they eaten something they shouldn't have, mixed with new animals, change of bedding, feed stuffs, weather?)
Is it something psychological? (Could they be upset, stressed because of changes in routine/living conditions)
Generally I know whether it's something that needs immediate treatment or something which should be closely monitored before calling out the vet.........but this only comes with experience - can't really learn it, other than by doing it

I'd suggest doing as much reading as you can - there are some great sites with forums that cover all the usual ailments, TAS is by far the best

but there are others too......The Pig site, the cattle site, and I'm sure there are lots of chicken, goat and others too.
Books are great as a reference (Amazon & eBay will have a HUGE selection on every topic you could ever want !) Some are expensive - The Merrick Vet manual for example is over £100 and although I don't own a copy I'm sure it's packed full of great info (but they do have a website too

) but each species will have books covering the anatomy and common ailments, along with general care and keeping advice.
And lastly, get out there and meet other keepers, help on other farms with experienced keepers and you'll learn bucketloads of helpful information. You'll also be able to learn how and where to inject or administer other treatments (stomach tubing, foot trimming etc etc etc)
Okay, so you'll not be a qualified vet able to write prescriptions or treat others animals, but you will have a greater understanding of how the animals function and instead of calling a vet out to visit every time you have a problem you'll be able to phone them and explain all the symptoms which will help them to identify it. It might not always be the case that you avoid a call out every time (and there are times when only a vet will do) but it all helps

HTH
Karen