Integration with G Suite
Choosing a webhost: 

Sounds like you are new to CiviCRM and you are starting a new site.  To your second - easier question - you can use any host for CiviCRM - if you have the budget - I would go with an expert that can implement everything to meet your needs on a VPS.  Otherwise if you are on a tight budget - these are two shared web hosts that I recommend (In No Particular Order):


CiviHosting - Supporting Partner of CiviCRM
Namecheap Stellar Business -  Not a partner of CiviCRM


I have used the above for different deployments and have been very happy. Alternatively - these are the requirements for CiviCRM, choose whatever hosting company you are comfortable or have experience with and make sure they meet the requirements for CiviCRM, any extensions you may use, and your CMS. In addition - I would recommend that you choose a web host that supports PHP imagemagick library. 

CiviCRM - GSuite Integration 

There has been a bit of work on this and I recommend you check these extensions out:


CiviGmail Integration with Gmail
Google Groups CiviCRM Integration
Google Apps Sync
CiviCRM - Google Script/Drive Integration
Integrating Email Client into CiviCRM
Inbound Email


Lastly - with all external integration - outlook, gmail, etc. - you should focus on getting your staff to use CiviCRM first. For example, don't place an item on your google calendar first - add it to Civi first - then you can have events assigned to you sync or be sent to your calendar. I am not sure what Google Drive integration would even look like - can't help you there.

Ill also say this - take baby steps - and you will learn how best to implement this for your organization. Maybe tight integration with G Suite isn't the best use of your time - maybe creating an interface in which Civi and G Suite are easily accessible from each other is an easier option. In the end - practice makes perfect.