Set or by the piece - you can order sushi either way
Sushi - you did it your way
If you're at the counter it's easy to see the case where the fresh fish is kept.
If you don't know the names of the fish just point at what you want to eat and the itamae (sushi chef) will make that for you. Normally, you'll get two pieces, but you can ask for just one piece also.
If you're at a table and don't know which fish you like best, or which you want to try, it may be best to order one of the sets.
Like all things in life, the more the cost, the better the fish. Sets usually have names like matsu (pine), or sakura (cherry blossom), or take (bamboo). The names don't really mean anything and one sushi shop's take may be another shop's matsu.
You might want to download an app on your iPhone that tells you what the English word is for many of the fish you'll see in a typical sushi shop. SushiGuru is one of the free apps and I've found it useful.
If you are ordering by the piece figure on spending more than you'll spend on the sets unless you are ordering top grade sets only. Some shops list the prices on little boards on the wall, but most don't, and even if they do they will almost always be in Japanese only.
Cost wise, ordering by the piece can be a few thousand yen - depends on how hungry you are and which fish you order - to several thousand (think 10 to 20 thousand) per person.
As in any restaurant alcohol adds to the cost, and can be a third of the total or more, depending on how many beers you consume or how much sake you drink. Sake is generally the drink of choice with sushi as beer can be too filling, but it's all personal choice, of course.