HShare is an interesting coin that is meant to be used as a sidechain for other cryptocurrencies. It's actually a dual chain that can be used for both block and blockless chains. This means that HSR has the potential to be very useful in the transfer of data between different chains, which for many cryptocurrencies is not currently possible.
This could potentially be a very valuable solution in the future where regulation and centralization begin to oppress traditional blockchain technologies.
It also features a dual purpose POS/POW model which both rewards token holders and also gives them the ability to vote for protocol updates and to determine the use of funds through a governance system that gives you a real role to play in your HShare investment.
You can, fortunately, buy HShare on most top-level exchanges. Your favorite cryptocurrency exchange is most likely represented. The most abundant trading pair for HSR is BTC so you'll need to purchase Bitcoin first from an exchange that accepts fiat dollars or your local currency in order to buy your HSR tokens.
When storing your tokens you have a few options, but using the official wallet is most likely the best option for you. This is due to the fact that if you do not use the official wallet you will not be able to participate in some of the prime advantages this coin brings you.
While they do have a wallet option available with HyperPay you will miss out on the ability to stake your coins for rewards and the ability to vote using governance should you decide to go this route. I'd recommend the desktop Windows wallet or the Mac wallet instead if you can. Be advised that they do not seem to have a Linux option available.