A lot of great advice in here. I'll just throw in my $.02, as I've acquired chickens at many stages of life.
Eggs
After many of my flock were massacred, I decided to hatch a few eggs from the chickens who had been killed so that I could continue their "line". I got a brooder online and hatched them in my bathroom. While it was really fun to go through the 21 days with them, I was also very anxious. I think that I made some errors in how I tried to keep the humidity, and only 2 of the eggs made it all the way. Watching them hatch was amazing.
Chicks
I raised the chicks that I hatched, but I also had the experience of raising chicks that were hatched in the PreK where I work and they needed a home for the chicks. Raising chicks is really fun. They develop so fast! One day they can hop up on a little box. The next day they suddenly have little tail-feathers! Raising chicks also means that you are mama--it makes it much easier to handle the chickens as they age. The major downside for me was that I would worry when I was at work--you read so many horror stories about how chicks can die or injure themselves. I personally spent a lot of time checking on them. Again, rewarding, but if you are emotional like I am or a bit of a nervous Nelly, very time consuming. Also, both batches of chicks were obviously not sexed, and so I worried that I might end up with a bunch of roosters. By luck of the draw I've never had more than two roosters at a time. I'm not sure what I would have done if 3 or 4 had been roosters. I'm a big softie and would not be able to cull an animal because of gender (or give one away knowing it was headed for the stewpot).
Pullets (older than 8 weeks)
I really liked the phase where my chickens were old enough to be out in the coop without additional heat, but not quite "adult ladies". What I think of as "the teenage years" were really fun. Watching them really learn to fly, establish their personalities, develop a pecking order, etc. They are much less dependent than chicks, but you still get some of that young animal bonding tome if you are good about handling them and interacting with them each day. If they were not handled before they got to you, they might be more skittish and if might take some time and patience to get them to let you easily pick you up or to come running over to you. Based on what you wrote in your intro, I think pullets might be a good fit for you.
Adult Chickens
Early in my chicken-owning adventure, I ended up getting a chicken from a local farm. We'd had a dog attack and our one survivor was lonely. The lady hen we ended up with was great. She laid for about a year after we got her, then her production slowed down. For the last year that I had her, she didn't lay a single egg. HOWEVER, she had more common sense than all of the other chickens put together. She was vigilant, cautious, and made sure that the hens stayed safe. She just made good decisions. Because I enjoy animal personalities, I don't mind feeding a hen who isn't laying. I do think that if you buy adult birds, you should make sure that you know how old the birds are. If you purchased 3 year-old birds, your whole flock wouldn't have a ton of time left, and you'd have to be thinking about how to add younger birds to the flock.
I do think it sounds like pullets are the best fit for you. However, later on down the line you might be into the idea of hatching your own eggs or raising some young chicks. I think that chickens have their own charms (and stressors!) in all phases of their lives.
Good luck with your flock!