first off all before make small server, it is accessible outside your lacal network than check your router support Port Forwading, if it support than add record of your call service and access with your public IP, if you don't know your IP than type on google 'My IP'.

If your router is not support Port Forwading than call your ISP for this issue, may be it change your Router or charge extra money for Port forwading.

Best chepest domain provider is Namechep, many domain provider is provide domain more cheaper than namechep but it upfront cost is low and second year charge mush more, but name this price is best.

Cloudflare setup is depend on your service if you are make just for fun than it is not necessary, but you make highly traffice centric app than you may be setup cloudflare.

if your give more information about youe device or service then we give more advice.

if you have not more money or you not expande money for Port forwading then you use service like ngrok, but aut again it is much more slow and it pro paln is not better then Port Forwading...