I currently have a lovebird, and prior to him I had a cockatiel that lived to be 22 years old. And I had parakeets when I was much, much younger.
I recommend a lovebird. A single bird, hand-fed, is a wonderful pet. They are very entertaining and have a "big bird" personality. My little lovebird is every bit as "parrot-y" as my Amazon. He loves to play with his toys and is just ready to take on the world every day. He is two years old now, 2.5 to be precise, and he has stayed tame and gentle. He likes everyone.
My cockatiel was a wonderful pet as well. He was also hand-fed and stayed sweet and loving his whole life. He loved everyone and would happily sit on a shoulder for hours, preening hair and nibbling on ear lobes.
Even though most conures are smaller, they sometimes don't make good pets for first time parrot owners. They can be loud. Really, really loud. And bite-y. I worked with a parrot rescue in Utah regarding a cockatoo that had been abandoned here, and ended up having some fascinating conversations about different types of commonly kept parrots. They told me that they get more requests to take in conures than any other type of parrot. People buy them because they are mostly small (those Patagonian conures are pretty big, though), and relatively easy to find. But they can be difficult. Like a lot of parrots, they can get fixated on one person. That can spiral out of control quickly.
Regarding cages: the rule of thumb is that you provide the biggest cage you can accommodate in your home and afford. Bigger is always better. However, you also need to pay attention to bar spacing. Bigger cages often have wider bar spacing and smaller birds can break their necks by sticking their heads in between the bars and then panicking. The ensuing struggle breaks their neck.
I love my birds. I won't get any more big birds. This little lovebird gives me everything I like in a much smaller, more manageable package!