You're going to spend a lot of time looking at your phone's screen, so make sure it's a nice one. People generally consider AMOLED screens to be the gold standard for smartphones, thanks to their vibrant colors, deep blacks and wide viewing angles. In certain situations -- say, when you're reading an e-book with white text on a black background -- AMOLEDs actually prolong your battery life because those dark sections aren't displaying the color black. Instead, the pixels are completely turned off. Meanwhile, LCD screens are more common on cheaper devices because they're cheaper to produce, but that doesn't mean they're awful. In fact, because the whole panel is backlit, they're often easier to see under direct sunlight.
You've no doubt noticed that smartphone screens are getting bigger, but that doesn't necessarily mean phones themselves are becoming unwieldy. Screens with an 18:9 aspect ratio are now more common, and they essentially allow you to view more content without having to scroll around as much. Even better, since these screens are comparatively tall, the phone bodies themselves generally aren't any wider than usual.
They aren't necessarily more expensive than phones with more traditional screens either: Devices like the Alcatel 1X give you an 18:9 display for just over $100. With all that said, if you have smaller hands, a phone with a smaller display -- say, 5.3 inches diagonal or less -- is obviously going to feel more comfortable. Smartphone makers continually try to squeeze the biggest screens they can into their devices, but there are still plenty of solid options out there that won't cramp your hands.