Below are some alternatives I've compiled that should help you shake off that feeling that Google completely owns your life. Before continuing, though, let me offer this caveat: the reason so many people almost exclusively use Google Apps for nearly every facet of online activity is because Google makes really good apps. That said, some of these alternatives might not be on par with, say, Gmail or Google+, so make a measured decision on what you need from these types of services and what features you can do without.
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With Scroogle down for the count, the two viable not-Google contenders to take its place are DuckDuckGo and Gibiru. Both sites are pro-privacy and ensure users' searches are encrypted by concealing your IP address from your search query. With either of these two search tools, your results will be the same as the basic results you get from Google.
Web-based Email - "The undiscovered country makes us rather bear those ills we have,
Than fly to others that we know not of." While Hamlet didn't have something as trifling as Gmail in mind when he said this, consider the sentiment's application when considering ditching your email. Your best free online alternative is likely Hotmail, but that service is owned by Microsoft. Fleeing Google for the cold embrace of Microsoft/Windows seems to belie any intent to emancipate and protect yourself from the corporate Eye of Sauron that you're trying to avoid.The other thing is: Gmail's really nice.
Another alternative you may consider is ZoHo mail, but with that you're going to have less storage with a free account (you'll have to pay in order to get more than 5GB). If you really want to spend some time weighing your alternatives, Wikipedia has a table comparing all of the more sought-after features for webmail services that might hasten your task.
Social Networking - I can't recommend Diaspora enough as far as privacy goes, but the average Google+ user will have a hard time getting through the door as its still in Alpha and therefore new accounts are invite-only. The obvious and immediate alternative is Facebook but, similar to how I explained the dilemma with free email hosting above, you'd basically be trading one poison for another. Given you're probably already on Facebook, and how nobody really seems to be adopting to Google+ that enthusiastically, this is one case where, if you must belong to a social network right now, stay with Facebook. Until Diaspora goes public.
Image sharing/hosting - Flickr. Flickr, Flickr, Flickr. There's not really anything that can be said about Flickr that hasn't been said before. It's a great service, offers content protection for users, and just recently launched a savvy new look to users' contacts. It incorporates the social aspect of photo-sharing and has a great user interface. Even if you're not looking to ditch Google Picasa with all of this privacy hullabaloo, I still recommend giving Flickr a look. You may find that you outright prefer that service to Picasa and the less Google in your life at this point, the better.
Blogging - WordPress is likely to be your best alternative to Google's Blogger. It offers up a comparable assortment of different themes for users to personalize the design of their blogs, you can host your WordPress blog on your own server if you feel so inclined, and there are a host of add-ons you can apply to your blog. Tumblr might be a close second if you prefer a deeper social media aspect to your blogging, or if you lean towards brevity when it comes to composing your blog posts.
Browser - Firefox or Opera are going to be the two non-Google browsers named as the preferred alternatives to Google's Chrome. Depending on whether you're a simple check-the-emails-and-maybe-Facebook user or a "power user," the different resources offered by the two browsers should accomodate most people looking to unmoor themselves from Chrome. Firefox might be more familiar to casual users while Opera will likely make power users wild-eyed with excitement.
RSS Reader - NetVibes could be your best non-Google alternative but, unfortunately, you're not going to have the complete array of features that Google Reader has. If you're dependent on tagging articles you like or even being able to search your RSS feeds, that won't be available to you with the free version. However, if those features aren't all that necessary to your RSS experience then it might be worth your time to take a look at it.
Cloud storage - Dropbox is probably a service you've already heard of when it comes to cloud storage. Granted, Google's GDrive was just announced recently so people have likely not begun migrating to Google's cloud service yet, but Dropbox has worked great thus far. And you know what they say about things that ain't broke.
If you're a really dedicated anti-Google centurion, you could probably live an online life free of any of their apps if you don't mind sacrificing some of the amenities offered from the Google World. However, keep in mind that you don't have to be a tee totaler just to keep your information safer. Using Google apps isn't completely bad - as mentioned above, some of their services really are probably the best you're going to find for free - but lessening your dependency on the Google brand as a whole might serve you well.
And as always, if any readers out there have alternative suggestions for any of the services listed above or something not covered (Docs alternatives?), feel free to add your piece below in the comments.