I am finishing a new site right now. In your expert opinion, should I get a lot of backlinks before asking google to index my site for a better serp, or does it even matter?
I wouldn't get a lot of backlinks in a short period of time. Build your link campaign slowly and everything will be fine.
You do not have to submit your site to google or ask google to index your site that method is outdated. If you build some high quality backlinks google will index your site much faster.