First, the good news: you don’t need Google to have a web presence or an Android device.
The bad news: many, many things are much less convenient without a Google account.
My Google-free existence involves mostly using my self-hosted domain. It was pretty easy to shift to my own email, using Thunderbird on my laptop, K9mail on my phone, and my ISP’s webmail interface everywhere else (Bluehost offers several interfaces, but I find that I like Roundcube the best). For phone apps, I use the Amazon app store instead of Google’s, as well as a few offerings from f-Droid. I know Amazon tracks what I install, but it at least compartmentalizes my data more: Amazon knows a lot less about what I do online than Google does. I use dotcal.com and aCal to manage my calendar. Between the advanced category excluder plug-in and the WordPress Android app, the Snapshots category of my blog does a great job of letting my post snapshots from my phone to twitter without clogging up my front page with every picture I send out. I use Sparse RSS to keep up with feeds on my phone. I’ve found a way to do most of what I used my phone and Google’s web tools for without having a Google account.
Here’s the thing, though, much of what I do is much less convenient now. I exported my contacts from Gmail and imported them to Thunderbird, but I don’t have them on my phone. (I’m sure there’s a way to import them to K9mail, but they still wouldn’t be synchronized.) My RSS reader on the phone doesn’t sync read items with the web-based reader I use. Even though Amazon has a number of apps, they don’t have as many as Google (and without a Google account, I can’t buy the Mighty Eagle for my kid to use in Angry Birds). Whenever Flash releases a new version, my phone is rendered obsolete until I can find a copy of the apk file posted somewhere online. Ephemeral video? Say what you will about HTML5, I still can’t get videos to stream from my hosting account, never mind the fact that my phone records video in h264 but won’t play it in a browser. Google Hangouts offer a very convenient form of video conferencing, but without a Google account, they’re off-limits to me. (I’ve asked for a Big Blue Button installation at work, and even though I’ve been told, “We should be able to do that,” I can’t seem to get anyone to actually make it happen.)
What next? My web and mobile usage are clunkier now than they were before, but I’ve gained a marginal increase in online/mobile privacy. How much? In terms of mobile, probably very little, given how much mobile providers track their customers. Is the slight increase in privacy worth the big decrease in convenience? I don’t know, but the Galaxy Nexus and Nexus 7 are starting to look awfully appealing.