Google has started releasing the migration tool to G Suite admins for G Suite domains on rapid release and the migration tool will be available for G Suite non-admin site owners from 19 June 2018.
Capabilities
To get an expectation of what's possible see
Classic Google Sites Deprecation & Migration Timeline but be aware it's only a simple migration tool and new Google Sites still does not offer all the features of classic Google Sites - see
Google Sites Classic Comparison. You can safely experiment with the migration tool to see what's possible now but you may find that converted site is not usable. This means you may want to wait until new Sites becomes more capable or consider looking for a different platform.
Migration Assitance
If you find that the migration may be good enough and want some advice on how to get over some pain points or want some help in getting a good design for your converted site please
contact us.