A little history here. Bubble drinks originated in Taiwan where they are immensely popular with young kids/adults. The craze traveled first to Canada, and then to the US, cropping up primarily in areas with a large popluation of ex-pat Chinese teetotalers who were craving the flavors of home. Thus, the highest concentration of Bubble Drink providers seem to be in SF, LA, Canada and WA. Bubble Drink establishments are not nearly as ubiquitous as Starbuck's, but you will have no trouble finding one in the Bay Area.
4 of the more popular and more widely located ones are listed below, in order of preference.

Why? 1) Variety, both of the kinds of tea and the fruit flavors. 2) Add-ins. Taking a cue for Japanese restaurants, where dishes are shown using beautiful lifelike plastic models, Tapioca Express has a nice plastic platter that showcases the various add-ins for your drink--big pearls, small pearls, coconut jelly, apple jelly. 3) Happy customer service. They are well-trained and explain what all the various add-ins are, in English as well as Chinese. Locations all over California and a few outposts in Washington, Texas and Canada.

The variety of tea and fruit flavors are similar to Tapioca Express, but I'm not sure that they have as many add-in varieties. The second place comes because their customer service can be uneven, and sometimes downright lacking. 3 locations (Cupertino, Milpitas and Santana Row). There was another unrelated Fantasia Tea in Columbus OH, that appears to have closed.
Oh, how I want to like Quickly. There are many locations all over the Bay Area, their popcorn chicken is yummy, and their milk tea is pretty good. But unfortunately, I can't get past the underdeveloped customer service and the non-payoff of its name. I've been to a few Quicklys, and I always seem to end up feeling like my order is a huge, complex imposition on an employee's otherwise peachy-keen day.
Now my pet peeve--the name Quickly. This implies fast food--McDonald's, KFC, or In-n-Out (which can be In-a-long-line-n-Out, but I digress). "Quickly" is the phonetic equivalent to the Chinese name of this chain Kuai Ke Li. All good and fine, except Quickly...isn't. Unlike In-n-Out, where even if you are standing in line, you see employees bustling about trying to get you In-n-Out, there is not a lot of quick at Quickly. Despite this,I still go there grudgingly, primarily for the chicken.
Finally--Honeyberry. To be fair, Honeyberry is really more of yogurt and sweets store. This is also reflected in the milk tea. Way too sweet. However, Honeyberry is the only one of them all that takes credit cards.
This was by no means a scientific or endorsed assessment, so feel free to chime in with your favorites. I'm especially on the hunt for small or independent shops that I can try out.
Eat Well. Be Well.