Till the time you don't have lots of traffic you can try cloud/shared hosting.
But if traffic is getting very high and you face downtime/complaints, switch to VPS. Go for Managed VPS if you aren't technically sound to setup a Server. A good VPS and optimized website can handle 7-8K Traffic at a time. If you are again breaking boundaries, Increase one more VPS and separate your Data and Database on different VPS.
At last, if it's really very heavy traffic and lots data needed to be stored, then go for dedicated servers.
Still, if you have any doubt post here.