I've seen this question on the internet a few times, and will do my best to answer it here. I know each person is different, so what's best for somebody might not be best for someone else. For this overview I will only include the series that are freely and legally available online, which limits us to Ultraman, Ultraman 80, Ultraman Max, and Ultraman Mebius. (the first is available on Hulu, and the others on Crunchyroll. For reasons, I'm not including Ultraman: Towards the Future.)
Side not: the best starter series is likely either Ultraman or Ultraman Tiga. However, there are many other valid options!
Most fans will tell you to start with Mebius. While I love Mebius, I have to disagree. Every Heisei series from Max to Ginga S has been made with the nostalgia factor in mind, but Mebius probably most of all. Not only is this series best enjoyed when you have seen all the previous Showa era series, but it contains some fairly major spoilers for both Leo and Ace. (I'd say it has spoilers for Taro, but he died so many times in his series they don't really count).
Out of the two available Heisei (that is, post 1989), Max is the best to start with. It contains many references to previous series, but not so many as to hinder enjoyment. Several of its episodes go back to Ultraman's roots: science fiction mystery, but it retains the super-hero element (which became fully introduced in Ace). However, it has no overreaching plot, no main cast of antagonists. I honestly would advise you to start with the showa era, but Max is a decent first series.
As the original Ultraman is available online, it's a very good choice. While it is not the first entry into the Ultra Series (that, for the curious, is Ultra Q), it introduces the giant hero the name has come to mean. It's a science-fiction mystery show ending with a giant monster fight. Certainly a very good place to start.
Please don't start with Leo. Why? It's in continuity with the rest of the Showa era, and so is best watched after you have at least Seven under your belt. Also, it alternates between extremely grimdark and super cheesy and weird. It's really gloomy at parts, and does contain more monster gore and human violence than your average ultra series. Still, it has an overreaching plot and villains, and if that's your thing, watch it before Mebius.
80 is another good starting point. It returns to the roots of sci-fi mystery, and while the episodic plots are seldom original they are usually well done. My biggest peeve with it is how it tried so hard to be feminist but ultimately failed. So close... but so far. Watch it before Mebius and you'll enjoy the 80 episodes of Mebi more, and you might cry over them.
Here's a short overview of a few other good starter series which are not legally available online: