Cloud comes in varying degrees. Starting from colocating your gear in someone else's racks to IaaS, PaaS, and finally SaaS. There are varying degrees in the *aaS offerings depending on what level of fault tolerance you are looking for.
The higher you go up in the offerings, the more you relinquish control and responsibility but gives you the most bang for your buck.
The biggest mistake companies can do when moving to the cloud is to lift and shift their severs and services 1:1. This is the most expensive option and offers the least value to most.
Let cloud providers take over handle services that are basically commodities these days (basic web, email, DNS, etc.) so you can focus on more important things.