For cryptocurrency wallets, you have four choices:
Web-based wallet: Which is what you have with Coinbase; they have the keys for you and make things easier for you (as you don't have to install anything locally). Coinbase does have the option of a "Vault" which adds more control back to you if you're concerned about them handling your private keys, though it's currently only for Bitcoin assets, so not useful for your Ether.
Full Local wallet: That would be like the Mist wallet/browser; you install it locally and interact with it on your desktop. It's more secure since you have the keys locally, and you get a copy of the blockchain locally, so you can interact with it directly. Downside is that the blockchain does take up a fair chunk of hard drive space.
Lite Local wallet: You don't need to have the whole blockchain just to transfer Ether around. Installing a lite client like Jaxx would allow you to transfer your funds from Coinbase to your local wallet, but you're then trusting Jaxx's Ethereum nodes to communicate with. You have more control over your private key, but it's still trusting a third-party.
Hardware wallet: A TREZOR- or Ledger-type device is the most secure in terms of keeping your private key safe, though you then need to use their applications or a web-based wallet that can interact with them (like MyEtherWallet.com) to use your funds.
So, if you're concerned about Coinbase having your private keys, you can either pick another web-based wallet provider that you trust more (generate a wallet on MyEtherWallet, or install MetaMask), or use a local client, or purchase a hardware wallet.
My general recommendation is to use different wallets when the value of what you're saving increases. Aside from the advice to never invest more than you could risk losing, I'd say using a local wallet (lite wallet app on phone or laptop) for "pocket change" daily spending currency, and a hardware wallet for long-term savings is a good route to go. Web-based wallets are okay for special uses (exchanges, etc.), but shouldn't be storing the majority of your funds.
Best Answer
If Coinbase said they sent a transaction, they probably did that. Probably your local wallet isn't completely syncronized and that is why you don't see your balance. Use an online explorer (i.e. etherscan.io) to check your real balance.