1) Get a paper one. Don't rely on a phone app for all your davening needs. For one, you'll want a prayer book you can use on Shabbat and holidays when your phone would not be allowed. There's just something about a physical prayer book. Your intention, your kavanah, will suffer if you daven regularly from your phone. An app or website should only be used when necessary. You and this book will forge a relationship through good times and bad. Or maybe I'm the only person weird enough to think you can form relationships with books.
2) Ideally, use the same siddur your synagogue uses. It's easier to learn one format, one order, one translation. Different nusachs have slightly different orders and translation. Even within a type of siddur (especially Artscroll), each edition may have a slightly different translation. That probably doesn't matter to your average Frum-from-Birther, but if you daven regularly in English, you will eventually memorize that English, and a slightly different translation will trip you up and slow you down. You may prefer a different siddur than your synagogue's, but I believe your life will be easier (and you will learn faster) if you use the same one as your synagogue. YMMV. As you become more fluent in Hebrew prayer, you'll be able to move between siddurim with ease. My advice: wait on that siddur with the slightly better translation or prettier font. Your needs and desires for your siddur may change over time as you improve.
But what are your options? Here are the ones I know; please comment below to let me know if I missed any!
Don't know the difference between Ashkenaz, Sefardi, and Sfard? Start here!
Artscroll: Let's not fool ourselves. If you're an English speaker, you're probably going to end up with an Artscroll Ashkenaz edition. Artscroll is the PC of the siddur world, for good or for bad. Most shuls have the older (but totally usable) edition, but you may want the new Wasserman edition. The only practical difference between the two is slight differences in translation and the Wasserman has slightly easier-to-read font. Nothing major is different in the translation, but it can give you pause when you have the English of one memorized.
It comes in Pocket Size. Let's take a minute to talk about pocket size siddurs. They're great for carrying in your purse or backpack, but you will want to consider how to minimize the damage from frequent travel. I prefer the hardcover edition, and a friend suggests placing the pocket size in a plastic sandwich bag to prevent the book accidentally opening and damaging pages.
Koren: The Koren is the most popular siddur among my friends, and it's gaining a foothold in synagogues. However, not many shuls can afford to switch over all their siddurim. The biggest difference in the Koren is that the English text is on the right and the Hebrew on the left; that is the opposite of other siddurs. Also comes in Sepharad (Sephardi), Pocket Size, and its older edition. The Koren is particularly popular among religious Zionists because it includes many Israel-specific instructions and prayers. I bought my first Koren specifically for my first trip to Israel so I could take advantage of that information.
As you become comfortable davening entirely in Hebrew, I suggest the Koren Talpiot edition. It is only the Hebrew text, but with instructions in English. I don't know of any other siddur with that format, and I find it much easier to use Hebrew when the instructions are in English. It's easy to pronounce Hebrew when you don't have to understand what it says! (Of course, knowing what you're saying is always preferable and may sometimes be halachically required. Let's talk about that another day.)
Birnbaum: Boo hiss... stay away from the Birnbaum, English daveners! The Birnbaum edition is the old-school "Thou Makest Thine Face Shine Upon Thee" edition. (I just made up that phrase and in no way guarantee it is readable.) Every shul has a few Birnbaums laying around, and if you're really late, you may get stuck with one. If you daven in Hebrew, it's not a big deal to use the Birnbaum. But if you daven in English, you will fall into a bog of language. I didn't know it, but the Birnbaum has a fascinating history: it was originally intended to be used by both orthodox and conservative congregations!
Tehillat Hashem: This is the Chabad siddur. Since many Ashkenazim (also?) attend Chabad, note that there are slight liturgical differences from nusach Ashkenaz. If you attend a Chabad regularly, you should definitely buy this version. If you use an Artscroll at home and the Tehillat Hashem at shul, it'll be difficult to get an intuitive feel for the service. Also comes in Compact Size.
Metsudah: From what I can tell, the Metsudah is just a kind of interlinear siddur (see below). I don't believe it is attached to any particular group, but it is an orthodox siddur. There is an older version.
Siddur Sim Shalom: The siddur of the conservative movement.
Kol Haneshamah: The siddur of the reconstructionist movement.
Hillel: Even the student group Hillel has a siddur! I don't actually know whether this siddur is appropriate for the orthodox, but I'm told it's a great learning tool and an excellent choice for a "learners' minyan."
Bonus reading: The U.S. Military just came out with a new siddur that seeks to incorporate all the movements into one siddur: U.S. Soldiers Getting First New Siddur Since World War II.
But wait...there's more! You not only have to choose among the "brands" of siddurim, keep in mind that there are special editions you may want to add to your collection, now or later.
Interlinear: An interlinear siddur helps you learn what you're saying as you daven in Hebrew. It can be formatted in two ways: a) English words are printed directly below the Hebrew word (Artscroll) or b) short Hebrew phrases written beside their English translations on the same page (Metsudah). To my knowledge, every prayer text (siddur, Tehillim, etc) by Artscroll is available in an interlinear version.
Transliterated: This is when the Hebrew text is written in English letters, making it easier for an English speaker to daven in Hebrew. For example: "Shema Yisrael." Beware: transliteration has its place and can be incredibly useful, but it can also become a crutch. A transliterated siddur may also include an interlinear translation.
Weekday: For a smaller siddur to carry during the week, consider a weekday edition. Also comes in interlinear and transliterated. Currently, the Koren doesn't come in a Weekday version. Personally, I daven most often with an Artscroll Pocket Size Weekday siddur that I keep in my purse.
Shabbat and Festivals: As you can imagine, this siddur is limited to Shabbat and holidays. Also comes in interlinear and transliterated.
Machzor: A machzor is just a holiday-specific siddur. You can buy a machzor set that includes Rosh Hashanah, Yom Kippur, Sukkot, Pesach, and Shavuot. Koren has recently released machzorim for Rosh Hashanah and Yom Kippur, and people rave about them. While you can buy them separately, you can save some money by buying sets of two (RH and YK) or five. Available in different nusachs and interlinear, and all those in pocket size. They also come in many colors and cover materials. Additionally, there are other machzorim-like editions such as Tisha B'Av/Kinnot (Artscoll and Koren) and Slichot.
The Ohel Sarah: This siddur is for women and published by Artscroll. I do not like it, and I do not recommend it. It was my first siddur. But this siddur belongs in the idealized 1950s, where all a woman did was care for children and pray for more. I am particularly bothered that I didn't know what Tachanun was for over a year because it's simply...not...in...the...siddur. Do you have any idea how weird Tachanun is when your siddur says nothing about what's going on around you?? The page where Tachanun should appear has an annotation at the bottom: "The practice of saying Tachanun after the Shemoneh Esrei was never established as obligatory. Since it is based on minhag (customary practice), there are numerous exemptions to saying Tachanun on many days [what a poor way to say that]. Without a clear indication that women accepted this practice, it is presumed that they never did." So...we won't even bother printing it. Heck, let's stop printing Ma'ariv while we're at it! Don't waste your money. /rant.
Tehillim: While technically a book of the Tanach rather than a siddur, people often daven Tehillim from a dedicated Tehillim book (Artscroll and Chabad). Tehillim also come in interlinear, transliterated, and English-only editions. (My favorite is The Book of Psalms in Plain English.) Of course, Tehillim come in all sizes, including pocket size. There are literally dozens of editions of Tehillim with English translations; go to a Judaica store to browse them in person to find one that speaks to you. There is no shortage of Tehillim apps, but most are Hebrew-only.
Apps: As mentioned above, there is no shortage to Tehillim apps. Siddurim are harder to come by. The most popular seems to be the Siddur app by RustyBrick (maker of many other quality Jewish apps), but Chabad recently released Tehillat Hashem as an app. Amazingly, despite a great deal of effort, I can't find a website for the Chabad siddur app, even on their main website. Search for Siddur Tehillat Hashem directly in your app store of choice.
Websites: There are surprisingly few English translations of the siddur online. Most modern English translations are probably protected by copyright laws. However, you can check out The Open Siddur Project and The Transliterated Siddur (not a complete siddur). If you bentch after meals with transliteration, adding a bookmark on your phone to the bentching on The Transliterated Siddur is great on the go!
What do you daven with? How do you like it? What would you improve?