There are many ways to watch any version of Dragon Ball, and there are plenty of websites dedicated to episodes of both old and new anime.

One of the best ways to watch Dragon Ball is to use Funimation's website. Funimation is the company that makes the english dubs for many different animes, and is the most trusted for them among anime fans. They have Dragon Ball, Dragon Ball Z, Dragon Ball GT, and the Dragon Ball movies. There are some that you may have to pay for, but it depends on how recent the series is.

Another way to watch is using Crunchyroll. Crunchyroll provides subtitled episodes of many different animes, but they only have fairly recent ones, so it's not the place to go if you want to watch older episodes. The only series they have available is Dragon Ball Super, but you can get the newest episodes quickly and catch up if you've missed any. You do not have to pay for Crunchyroll, and the only significant benefits you get are no adds, and you can watch anime an hour after it airs in Japan. But, even though that is a benefit, it will be available to the non-paying public soon after.

Yet another way to watch Dragon Ball is on Hulu. Hulu is similar to Funimation in a few different ways. They have Dragon Ball, Dragon Ball Z, and all of the movies. Although it may not have a few episodes of Z, so that's when Funimation becomes more useful.

These are the three best ways to watch Dragon Ball, Crunchyroll for the new, Funimation and Hulu for the old.