TL;DR: use localbitcoins or coinbase to buy - store in a paper wallet.

Coinbase does certainly have its fair share of issues - it's the simplest to buy with though since you don't have to deal with buyers/sellers like you would on a traditional exchange.

If you're concerned about Coinbase then try localbitcoins. It's where I sell my bitcoins because coinbase doesn't allow me to sell in the UK.

The benefit of local bitcoins is that you have access to your private key (coinbase don't allow you to see this) but it does mean you are dealing with a bitcoin dealer rather than coinbase.

Either way, I wouldn't recommend storing your currency long-term on any of these.

You can store many different ways, the most secure way (what I use) is a paper wallet. You can download the wallet generator to your desktop and create a wallet offline without ever being connected to the internet, then print it out, send some bitcoin to the address and store the paper somewhere safe. The downside to this method is that your coins are only as safe as you're physical storage of the wallet. Think of it like storing cash.

I personally haven't tried a hardware wallet like Trezor - it is certainly intriguing and I might invest in the future. And then you have software wallets like Jaxx, Mycelium, and actual Bitcoin Core (which syncs the blockchain on your PC - useless unless you're a miner), these wallets are great but I find it difficult to trust the security of my phone/PC these days - hackers are pretty good at what they do - if you want the full security of blockchain you can't leave any holes. Which is part of the reason I think Coinbase is a better place to store if you're unsure, since they keep all the coins in air-gapped cold storage, and are insured against loss (however much stock you put into that claim).

Good luck!