I think basic common sense will keep most issues at bay for small flocks, especially if you are not constantly bringing in new birds. Keep food and water clean and pest-free (it's easier than you might think...keep food in galvanized trsh cans in a dry area, fill water daily, run feed and water pans through dishwasher weekly (?) or wash with soap and water as needed). Nip anything unusual in the bud. You will become a poop watcher as it is one of the first signs and easiest things to monitor for illness or parasites. It just requires a casual observation of the surroundings as you go about the daily visit to feed, water, collect eggs. Look at the birds daily, as you go about daily visits, so that you are familiar with how they look, how their feathers look, what they look like when they are healthy.
There was another thread recently where someone bought 5-6 chicks, lost a couple, had 2 that needed vet care int he first few months...If it were me I would re-home the rest and start all over again with chicks from a different source. Raising chickens shouldn't be that difficult.
We started with 6 RIR pullets at point of lay. I got them from a guy who posted on craigslist for free. He was frustrated with feeding them for 4-5 months and hadn't gotten a single egg from 6 girls. When I picked them up, not knowing zip about chickens (just thought over a period of time that if might be cool to have eggs, and something I had always wanted to do), I noticed that he had them housed in a chainlink dog kennel with an igloo doghouse for shelter and had BIG un-friendly dogs in kennels on either side of the chickens. Well, I proceeded to stuff the six pullets into a large dog crate and took them home to something my husband threw together with a roll of hog wire held to the ground with tent stakes. The kennel placed in the hog wire surround was what they lived in for a few weeks until I found an a-frame coop free for the taking, also on craigslist. Even in those meager conditions I started getting eggs from them within days. That was in September. I moved them into the A-frame before winter hit and continued to get 4-6 eggs per day all winter.
I happen to live in an area that isn't conducive to parasites (very dry in the summer, and yet gets below freeezing in the winter) so we haven't had any trouble with parasites or worms. But I do keep things clean...bowls, bedding, nests, roosts, food/water. I don't think keeping chickens is much different than taking care of my dogs...basically keep things clean and use common sense.
Raising chickens the easy way is very do-able. The only thing I suggest to people to make it easier is to acquire the coop first and then the chickens.