UPDATE (13/09/14): Added alternatives to G+ Photos and Drive.
The big list for alternatives to Google services (see here for the why). When writing this article, I realised just how pervasive Google has become in the massive range of services they offer. My criteria for alternatives was to be standards-based, preferably open-source and self-hosted, secure, have an easy backup/sync facility, a solid business model (I’ve purchased the product or subscription wherever possible to support this), Android presence, and not another company like Google. For example, it would be senseless to switch from GMail to Yahoo Mail for the same reasons you leave. Yes, I’m also aware that not all of the alternatives are self-hosted or open-source, but sometimes a slight compromise is necessary.
Condensed List (see below for explanations and links):
GMail Service => Fastmail (Subscription | On AWS) / Self-hosted [ie. Dovecot,Postfix]
GMail Client => Thunderbird (Desktop) / K9 Mail + APG (Android)
Google Calendar/Contacts => Fruux (Hosted in Germany) / Self-Hosted [ie. OwnCloud, Baikal] / Fastmail
Google Chrome => Firefox / Chromium
Google Search => Startpage, IXQuick, DuckDuckGo
Google Docs => Zoho / Self-hosted (OwnCloud, Kolab)
Google Drive Storage => Self-hosted (Bittorrent Sync, OwnCloud)
Google Reader => Self-hosted (Tiny Tiny RSS)
Google Play Music/Movies/TV => Self-hosted (Plex Media Server)
Google Authenticator => Authy, FreeOTP
Android Device Manager => Cerberus, Prey
Google Chromebooks => Second Hand Lenovo laptop / Chromebook with Ubuntu
Google Blogger => WordPress, Drupal, Joomla
Google Play Books => FBReader, Calibre, Mantano
Google Maps => OpenStreetMaps (Desktop) / OSMAnd+, City Maps 2 Go, Skrobbler, Maverick (Android)
Google+ Photos => Self-hosted (Album, OwnCloud, Plex Media Server)
Google Hangouts => Mumble, RTP
Android => Clean CyanogenMod or AOSP-based ROM, with Privacy Guard and XPrivacy module
GMail: The big kahuna. Either host your own (preferably on a redundant VPS), or use a managed and paid service like Fastmail. There’s a 60-day free trial, after which you pay approx $3 monthly fee for 15GB of storage, and you don’t get ads thrown at you or sold out like a chump. I’ve been using Fastmail since the start of the year and it’s fantastic, it’s what GMail was at the start – instant, responsive and simple. The spam filtering is as good as Gmail, and the rules/filters are even better (hello regex!). It has a one-click non-destructive auto-import for ALL your Gmail (my 160,000 messages/7GB took around 30 hours to transfer). It has useful keyboard shortcuts, also does IMAP correctly, so you use a mail client like Thunderbird for full sync/backups. Use MailDroid or K9 for Android, not as pretty as GMail, but they do push, and work well. Set up a catch-all forwarding rule in GMail and eventually phase it out. For added pizzazz, buy your own domain and use that email address – either use MX records or Fastmail’s DNS server. There’s another good guide here.
Google Calendar/Contacts: Use your own self-hosted calendar/contacts service (like Baikal), or a third-party service like Fruux (a German company, backend provided by AWS), free for small deployments. Calendar/Contacts uses nearly no resources, you could host it on a free AWS micro instance. These tie in with CardDAV/CalDAV for two-way sync between all your devices. Web front-end, Android clients (creatively named CardDAV/CalDav Sync) and desktop (Thunderbird/Lightning add-on). I prefer Business Calendar on Android myself as the calendar frontend.
Google Chrome: Chrome used to be fast, but now it’s a massive hog. Chrome also has a bunch of phone-home features like a leaky sieve. Chromium is better, but for depth of extensions, you can’t beat Firefox. The recently released 29 version even looks like Chrome. It has cross-device tab/bookmark/cookie sync, it’s fast and reliable and the Tree Style Tabs (which displays tabs in a list view) is a lifesaver for those who often have dozens of tabs open. Even the Android version allows ad-blocking/privacy addons. A must have. Chrome is still faster on heavy JS sites, but for a good cause, I’m willing to spare the cycles.
Google Search: If you’re logged in while searching, your Google search results are kept for a couple of months (even if you turn off web search history). Even if you’re not logged in, your searches can be correlated from your cookies and IP address to form a profile about you. If you’re used to Google Search syntax, you can use an anonymizing service like Startpage (also known as IXQuick). Alternatively, use a meta-search engine like DuckDuckGo, but I found the results lacking, or I’m just too used to searching like it’s Google.
Google Drive/Docs: I don’t use the cross-collaboration features of Docs often, but if you do, there’s Zoho or Kolab. For storage space. You could Truecrypt up a volume, but it doesn’t respond well to the delta-updates that Dropbox/BTSync uses, meaning you’ll end up re-uploading that entire big encrypted volume each time you make small changes. EncFS doesn’t have this issue since it’s file level. For Android access to your encrypted files, use Cryptonite. If you want straight mirroring between devices, then use BTSync (which has an Android client with folder sync), this is what I use. Alternatively, there’s also Seafile and Syncthing a pretty solid open-source cross-platform file sync.
Google Reader: Everybody needs a good feed reader. If you only peruse a handful of feeds, then you can always use Feedly or Flipboard. But for industrial strength or peace of mind, use Tiny Tiny RSS. It’s self-hosted, open-source, infinitely customisable with a full plugin system and you can even use regex matching for article filtering, which in turn can either be score-adjusted or filtered completely. Your choice of PostgreSQL/MySQL/other, and it’s very quick, especially over LAN. A massive time saver, and it’s been polished over 9 years by the creator.
Google Play Music/Movies/TV: If you don’t already use Plex, spend some time and set it up. It’s the best media library and cross-device player I’ve used, including media metadata, and even remembers your progress cross-device. It’s based on the open-source XBMC, and it works extremely well for managing your media library, it even does server-side transcoding, though this requires a fairly beefy setup. This also includes music streaming (complete with album artwork), without any limits.
Google Authenticator: For two-factor authenticators, the closed-source Google Authenticator used to be the standard. But now there’s Authy. It’s drop in compatible everywhere GA is (including scanning QR codes), it’s free and best of all, it’s open-source. Even the UI is miles ahead. I switched everything across within 5 minutes. There’s even a desktop version, though some may argue that defeats the purpose.
Android Device Manager: Cerberus was around long before Android Device Manager. It also does alot more, and it’s about $5 for a lifetime license. It works very well. Alternatively, you can check out the substantially more pricey Prey (which I also used on OSX/Ubuntu).
Blogger: It’s horrible, in trying to be overly fancy with JS both on desktop and mobile. It doesn’t even load AT ALL without scripts. Use WordPress/Drupal, or even better, self-hosted.
Android: Disable the phone home features like Location History and Web History. Don’t use Google Now and disable the Google Keyboard’s network function. Use a proper third-party launcher like Nova. Android runs perfectly fine without reliance on Google’s apps.
Google Play Books/Magazines: You should be holding all your books/magazines in .epub/.mobi formats, and you can stream them from Dropbox if you’re on the road. For reading, FBReader/Calibre will do it, but I haven’t found an Android client smoother than Mantano, it even does TTS.
Google Maps: There’s OpenStreetMaps, which draws on a huge amount of local knowledge, but it’s a few features behind Google Maps. On Android, two solid offline-capable alternatives with turn-by-turn navigation include OSMAnd+ and City Maps 2 Go. Alternatively, for a very slick navigation option, also check out Skrobbler (now owned by Telenav).
Google+ Photos: Switching to Flickr or another online service defeats the purpose. One possibility is a self-hosted front end gallery that supports local storage, like OwnCloud, which also does a huge swath of other things – Calendar, Music, Files, Notes, Contacts. However, I found Owncloud very flaky in use. Both Bittorrent Sync and Plex Media Server support auto-backup of camera uploads. I’ve since moved to hosting my pictures on my own web server, using auto-generated HTML galleries by Album (included in Ubuntu repos). It generates suitably sized thumbnails, optionally extracts EXIF information, rotates photos and even supports GPS data. Not as pretty as online services, but it gets the job done, and conserves bandwidth on mobile.
Google Hangouts: More or less the only promising part of Google+, but if you’re going to switch to Skype, you might as well stay on Hangouts. I can’t think of a viable alternative for videoconferencing, but I’m looking. For audio only, there’s Mumble.
Youtube: Just keep yourself logged out, take privacy precautions in your browser. You can keep track of your favourite channels through other means.