Simple answer: you're not going to find any other company that's significantly better than Google with regards to privacy. They're not perfect and they screw up (see: Google Buzz initial rollout or censorship of Blogger), but they're generally benevolent about the data they collect. Personally identifiable info is deleted in a somewhat reasonable time (though I do think it should be shorter), and flaws in e-mail privacy have more to do with antiquated laws than Google itself (data stored on servers for X time is legally considered abandoned and can be accessed without a subpoena - yes, that includes your 2 year old GMail contents).
Your best bet, if you don't trust Google or don't trust that our laws will eventually become reasonable with regards to technology, is to spread yourself out. You'll lose all the benefits of interoperability, but you won't have all your eggs in one basket.
GMail
Frankly, if there's only one Google app you use besides Search, it should be GMail. It's just that good. But, if you want to use some other free online e-mail service, there's always
Yahoo! Mail or
Hotmail. Alternatively, you can keep using your GMail address but get the data off Google's servers by using
Thunderbird and setting it to delete the e-mails on the server when it retrieves them.
Google Calendar
Mozilla also offers a
calendar product. Online, there's
30 Boxes and
Yahoo! Calendar.
Google Docs
Zoho is a very good alternative to Google Docs, and it turns out they have an e-mail service as well.
Google Reader
Netvibes is more than an RSS reader, it's "your personal dashboard," making it also a replacement for your iGoogle page.
Bloglines is another popular alternative.
Google Maps
Ask Maps,
Bing Maps (the only reason to ever go to Bing IMO, their bird's eye view is pretty cool)
Google Profile
Flavors.me,
dooid,
magntize
Beyond that, we're getting into apps that already have well-known alternatives (Flickr instead of Picasa, etc), so I'll stop there. Thing is, in most cases Google's apps are still the best. It's also worth noting that these other companies aren't going to be that much different than Google when it comes to storing your data. I find it easier to stay on top of one company's actions with my data rather than 10, and I like the integration Google products offer me. I understand why others may have a different viewpoint though. As cynthetiq says, the only real way to keep your data almost completely secure is to keep it off the internet altogether.