It may sound scary, but doing that first blockchain proof of concept (or experiment, as I think of it) really isn't. There are lots of ways to get going--from the proverbial "hello world" on TestRPC to much more sophisticated installations of your own Ethereum or Hyperledger servers.

As with anything, the hardest thing is to change inertia and step into the unknown. I think you need two key things to get going with your first blockchain experiment:

A use case. Something simple, but meaningful to you, that you would like the blockchain to prove. Tracking flow of data from one node to another? Verifying a settlement flow? Reporting on data aging? ...
A technical approach. Use a vendor hosted cloud blockchain platform? Stand up your own network? Choice of blockchain technology (e.g. Ethereum vs. Hyperledger)? Private or public network? Off-chain storage (what are the options)? ...

Ethereum is the more mature of the technology platforms and has, by far, the current largest development and support community. It's a good starting point to learn about implementing blockchain and how it would meet your use cases. I wouldn't stop there though. Learn from the first proof of concept and then ask more sophisticated questions (mainly of yourself!).

In the meantime, if you are ready to experiment with Ethereum, I think you need a checklist to get organized. Below is mine (with increasing degree of difficulty):

Install NodeJS
[Optional but needed if you want to use containers] Install Docker
Install TestRPC and learn the tool
Install Ethereum and do a Hello World
Install Truffle and learn the basics
Learn Solidity, primer on the language and its gotchas
Write a smart contract and compile it
Deploy a smart contract, consider migrations and chaining
Write automated testing for smart contracts and how to test transactions (Truffle console)
Write (or reuse existing) frontend Dapps
Integrate with a crypto wallet (Metamask works well)

Have fun experimenting :)