Isn't The Cloud just "running my stuff on someone else's hardware"?
In a word, NO! If you're in IT at any level, from individual contributor all the way up to CIO, you need to take the time to really look in-depth at the cloud offerings available today. If you aren't blown away by what's available, then you need to spend more time looking at the capabilities on offer. That's not meant as an insult - it's my honest opinion as a deeply technical consultant who has been in IT for a very long time. Personally, I would recommend that you look at AWS because they're the leader and have been since Cloud became a thing. They simply have more resources available (documents, videos, tutorials, use cases, case studies, third party tools, etc.) to really show you what they offer, and it's absolutely incredible.
OK, I'm impressed, but my IT department can provide everything we need
You may be getting everything you currently think you need, but I promise you there are more capabilities out there that you just haven't thought of yet because of limitations that exist in your current environment. For example, what reply would you get from this request:
I would like to see a topology graph of all of the server, network, database and security resources associated with Application X.
I've worked with thousands of companies of all sizes, and I've only seen this question answered a couple of times, and only for very well-known, small applications. The tools to answer it are out there, and many companies own several of them, but there are technical and political obstacles that prohibit this information from being displayed on a single pane of glass. However, with AWS, there are several third party vendors that offer products that can provide this information within minutes. Specifically, all of the components are registered centrally within AWS, so their metadata can be retrieved with the AWS API. These third party tools pull the data and display it on a graph to make it easier to consume (with filtering so you can include/exclude the appropriate components based on name or tag).
This central repository of configuration information is, basically, a built-in CMDB. There are companies that have been working for years and years to eventually have a partial CMDB, and the big cloud providers offer it from day one. And in AWS, this central repository is audited BY DEFAULT. That means you can see exactly who changed exactly what and when. That's incredible.
I still don't see what's so great about The Cloud
That means you still haven't spent enough time trying to understand what's available. What I would recommend is that you go through one of the AWS workshops available on Github. Specifically, go through the Website workshop available here:
It will step you through the creation of a serverless web application, including a user self-registration component. This is something that's normally a HUGE obstacle in enterprise application creation, and AWS offers it directly via their IAM service. And, as I mentioned, everything is defined centrally, so you can see what your applications look like.
Now I'm impressed, but it looks pretty difficult to set up
It is definitely complicated, but it can be done. You do need to define policies around things like naming, tags and usage, and you need to restrict who can perform which actions, and there are a multitude of other policies that must be defined for your specific enterprise. The good news is that there are several education and certification tracks available to get people certified as AWS architects. Additionally, there are lots of AWS architects available for hire. It's like any new initiative - it just needs to be approached incrementally.