I see. A lot of older folks have some trouble with phones. Luckily, there are many to choose from and even more guides and people around to help!
If you are looking for a bigger screen, most modern Android phones have at least a 4.5 inch screen. This is a nice size increase from the iPhone 4s' 3.5 inch screen. With a screen this big, you want to look for a screen with at least a 720p resolution. Once you hit five inches, you should be looking for a 1080p screen.
The iPhone 5s has a four inch screen and is a nice phone, if you want to keep him on iOS. The price is rather high, though.
If you want to go Android, you should look at the Moto G with LTE. It is a nice phone with great battery life, a nice 4.5 inch, 720p screen and is a mid range phone. This phone costs $220, and is unlocked.
The next step up would be the Moto X. It has some great features, good battery life, a 4.7 inch 720p screen and is a flagship device. I could also suggest the Nexus 5, which is a great phone. It has good battery life, Vanilla Android, prompt software updates, and a 5 inch 1080p screen. It is Google's flagship as of right now. Both of these phones cost $350.
After these phones, it gets a little more complicated. There are many phones, and prices are about what the iPhone 5s currently costs.
Are you looking for a phone with a new contract, or an unlocked phone?