Before moving to Cloudflare, is your Website working over HTTPS connection?
There is a way to add missing DNS records or import them directly before you switch domain nameservers to Cloudflare (you add domain to your Cloudflare account, then wait a day or so, in the meantime compare and modify settings and DNS records, when ready, change domain nameservers).
Sharing some helpful articles about managing DNS records at Cloudflare dashboard:
Quick tip: Good practice is to check and disableDNSSEC at domain registrar if it is enabled, before switching domain nameservers in general.
Furthermore, regarding subdomains, some issue were but mostly related to sub-domains which didn’t had an valid SSL certificate before moving to Cloudflare - also related to main (naked) domain and similar like www, mail, etc. I’d suggest you to take a look at the article from below about SSL for sub-domains:
Nevertheless, there is a known solution if you have deep sub-domain case like def.abc.domain.com and similar. Kindly, I’d suggest to check below article for a solution if so:
I remember there were some topics about issues with GoDaddy hosting / SSL certificate, might have to try and use search icon as far as for some customers, GoDaddy didn’t incluced an SSL certificate at all:
Here is a way to re-check if you correctly setup the SSL for your domain with Cloudflare:
Regarding available SSL options at Cloudflare dashboard, check here:
If any other issues appear, follow the needed steps for troubleshooting from article below:
In case you do not have an SSL certificate, you can use Cloudflare SSL, if so, kindly make sure you follow the instructions as follows on the below article to setup an SSL certificate using Cloudflare Origin CA Certificate: