Yes, it is possible to see and sync your contacts to and from Google. The best way to handle this IMHO, is to keep all of your contacts in Google. While it isn't necessary, it does keep things simpler. The first thing to do is go into the settings on your phone.
Settings/Accounts/Google - Is the account that contains your contacts listed? If not, add it. Next tap on that account and go through the
Sync Settings. One of the lines should read
Sync Contacts - make sure it is set to ON. You might want to check the rest of the settings while you are there too. I don't use it, but you might want
Sync People Details on too. I'm not sure if that only applies to Gmail, or Contacts too. When you are done exit your
Phone Settings and go into your
Contacts app
Settings. On my Samsung Note 5 stock Contacts app I am only allowed to choose 1 account. Yours may allow more. If you want all your contacts in Google select
Import/Export Contacts and export your contacts to
vCards. You will be able to use those to import into your Google account. Next go back to the previous screen where you will see a line
Contacts To Display - make sure that your Google account is listed, and add it if it isn't there. Then exit
Settings and confirm that your Google contacts are now visible. If you are able to see your Google contacts and your Phone contacts you will also have the option of merging the data so that you only see 1 entry for each contact. If you can only see the contacts from one account, or you want all of your contacts in Google, go back to
Contacts/Settings and go back to
Import/Export Contacts. Now select
Import and you will be able to import the contacts previously exported from your phone. At this point you should be able to see your contacts on any device that you own which has access to Google (some set-up may be required). I hope this helps.