I break down the 4 policy tree choices at the start as follows:
80% - Tradition: 8 free buildings is a huge boost (4 aquaducts, 4 cultural buildings), and the happiness boosts in Tradition are usually the best of the early SP trees, since you get extra happiness in the capital per every 2 citizens and extra happiness per every 10 citizens in every city. Moreover, you get a food boost in the capital and extra gold there, too, which is vital in order to have internal trade routes early on before linking all your cities with roads.
15% - Liberty: A free Great Person with the closer is still fairly handy for grabbing wonders out of the later portion of the early game, or even for founding a religion in some spots. The free settler is helpful in isolated starts. Often, I go Liberty when I find myself on a small continent alone or when I know I need a viable early wonder (GL is never possible, but usually by Chichen Itza I can rely on the GE from Liberty for it). Also, the extra production per city can be useful right away in jungle starts.
~4.5% - Honor: I rarely open Honor first unless it's just for the opener, which I only do when I know a lot of open land will be around for some time in which barbs will spawn. Generally, I only go all the way through Honor as my first tree when I am a civ with a very early melee UU (Greece, Rome, Persia) or a very powerful melee UU that occurs later (Impi, Samurai, Berserker). The main problem with Honor is that it doesn't boost happiness much at all now (IIRC, it used to include happiness either per wall/castle or barracks/armory, which got moved to Autocracy).
<.5% - Piety: Opening Piety and going all the way through is something I've done once in BNW (out of maybe 20 games ranging from brief Duel games to Standard-sized normal ones). The only time I opened it, I was playing Poland and started very close to a faith mountain. Because I knew I could recover if going Piety wasn't useful via Poland's UA, I went Piety and worked the mountain and succeeded in hard-building Stonehenge. I then bee-lined to a Reformation and took Jesuit Education. This was the only time I opened Piety, and it worked fairly well. But this was highly, highly unusual. Opening Piety is something that I'd only see doing in situations like this, where I know I'll have both a cultural boost (Poland's UA) and a faith boost (faith-yielding natural wonder) in order to both found a religion as well as quickly start on Tradition or Liberty as my 2nd SP tree. Otherwise, Piety is the worst to open, as it provides zero happiness and doesn't directly aid production, food, or expansion whatsoever. In general, I tend to open and go through Piety as no earlier than my 2nd SP tree, since I often won't have a religion and/or high enough faith yield until completing my 1st SP tree anyways. Piety is a good tree, and a great one to back-track to, but a terrible one to open.