Common site vulnerablities and how to prevent them during development of a website for a client?

Validate all inputs. Validate stuff on server. Don't use technology that isn't maintained security wise.

XSS, html escape everything while storing or presenting.. validate all forms .. read the owasp guide to prevent such things.. pretty much standard if you are using any good framework which takes care of this for you.

Just using Dev rant for what it is meant to used.

sql injection (similar attacks are possible with meteor/mongo if you don't check params to methods)
shit password storage algorithms (try not to roll your own authentication in general)
shit admin passwords
not enforcing ssl (more a problem for the users)
script injection
exposing secret api keys
no or wide-open spf/dkim/dmarc policies, enabling phishing
server admin stuff

Hiring developers who can use google

Never trust client.

Hit at least staging, if possible Dev too, with w3af, zap or something similar. Don't go to production while it still finds something... That won't replace a proper pentest, but the absolutely basic and obvious security issues will be identified.