I agree wholeheartedly with Boots. Riding pens would be a great place to start and most feedlots have a high turnover rate so they are almost always hiring. My brother worked in one for almost 10 years and I've done a fair amount of daywork in a couple nearby lots.
Like Boots said, you learn how to spot a sick one and how to tell them from a chronic (a cow that has bad lungs or something that can't be treated) and how to treat whatever sickness they have. Most the lots around here have their cowboys spend time in the vet clinic on-site to learn what drugs treat what and how much each cow should get...and how often. Mostly learning basic vet treatment for common ailments.
Working in there, you'll learn how to handle cattle, both sorting and driving/gathering, and how to keep things cool. IMHO, it's a great place to work if you enjoy that type of work and are willing to work hard.
The cons, as Boots said, it doesn't wait for the weather to break. You're riding pens when it's 20 below zero and blowing snow and when it's 115 and the wind is blowing 60 mph. Horses and humans alike tend to be prone to sinus and respiratory tract infections due to all the dust they breathe in all the time. The hours can be very long. My brother worked in a very large lot and he would average 15-16 hours a day. I worked in mostly small or starter lots and would average 12-13 but would occasionally get to 15. You're starting as soon as it's light enough to see (sometimes before if you're shipping...that gets done by flashlight LOL) and you don't stop until you're done, no matter how long it takes.
BUT, all that work can turn out a dang good cowboy with some time and in 30 days, you'll have a very broke, handy, and savvy horse. You learn a lot and you get really good at the job really quick if you listen and learn.