The barrel forms the main part of a dart. When we are speaking of "buying a (new) dart" we mean getting a new dart barrel. Nowadays barrels are available in almost every possible shape, with a wide selection of knurl or rings on the surface for a comfortable and safe grip.
The main barrel shapes are:
Some dart barrel shapes. The selection shows barrels with 2 threads (soft tip).
It's not possible to say which one is the "right" one. Based on a very theoretical reasoning the long cylindrical barrel could be the best of them because it is the slimmest and allows the tightest grouping. Good for 140s and 180s. But this reasoning doesn't help you when you are not able to handle such a barrel. Selecting the right shape is a matter of personal preference. Choose what feels comfortable for you. This is much more important than an abstract or "academic" reasoning about theoretical advantages or disadvantages of some shapes.
The surface of the dart barrel is covered by knurling or rings for a safe grip on the dart. Again there is a vast selection of different surface shapes, and combinations of them. And there is the possibility of a plain surface (which beginners are usually unhappy with). Again, it can't be said which variations are the best ones. Choose what suits you, what feels good and what you think you can handle.
There is only one thing you should avoid: painted surfaces. Paint might look good, but usually it makes the dart only more slippery. The probably only exception here is Unicorn's "gold" brand. These darts are covered with a golden coat (I don't know what material this coating is. But it's not gold) which they say improves the grip. I have not experienced any differences with these darts, but some people seem to like them. Try and find out for yourself.