From what I can tell, it should be possible to get some progress on this fairly easily.
First things first, make sure you've updated to Windows 8.1 This is essential, as without it you won't be able to update to the latest version of Mail, Calendar and People.
Second, make sure you're up to date with the latest Windows apps from the store. Go to the store and check for updates if necessary.
For Mail, just go ahead and try adding an account - you should be able to try adding a Google account, but if this doesn't work, you can also try IMAP. Instructions from Google are here:
https://support.google.com/mail/troubleshooter/1668960?hl=en
For Contacts, again, just try adding the google account and see how far you can get with this. I've not used it myself but it's worth a go.
Calendar seems to be the only app where you can't explicitly select 'Google' when adding an account. There is one way you could try to work around this limitation. If you have a Windows Live ID, then you essentially already have an Outlook.com account too. I know I know - you don't want to migrate to these services - but you can use it as a 'go between'. First, go to Google and publish a link to your calendar. Then take that link, go to Outlook.com, switch to the calendar and add a published calendar - use the Google URL you got. Once done, when you sync your Microsoft account to your Surface, your Google calendar should now show up.
The limitation is that you may not be able to /add/ events, only view them.
If you pin the Google Calendar to your start screen however, then this can be your 'go to' place for adding appointments, and all you'd really need the built in app to do is give you reminders etc (if that's what you'd be gunning for).
Let us know how you get on. Google haven't exactly been sticking with their 'don't be evil' mantra when it comes to supporting Windows devices, but it's worth noting that most vendors (Microsoft, Google, Apple) deliver the
best experience to their own platforms first! I switched from Gmail to Microsoft and couldn't be happier, but I understand you might be wanting to stick with Google. Good luck!