Hi, Welcome to Bitcoin family. Coinbase is quite old and trusted for storing your bitcoin. However, if you want to store really high amount of bitcoin, then I will suggest you to use a desktop wallet. But a desktop wallet will take a long time to get synchronized at the first use. So, for small faucet payments, Coinbase is good.
Now, once you have a bitcoin address in coinbase, you can visit faucetbox.com. It is a micro payment wallet and they have an extensive list of faucets. The small payments from faucets will be stored at faucetbox account and you are free to withdraw the payment in to your coinbase wallet as and when you want. Please note, that you don't need to create any account with faucetbox. Just use your coinbase address.
Please don't recommend storing large or any amount of bitcoins for that matter on a desktop wallet, especially to newbies.
If OP needs to store large amounts then he should probably just use the Coinbase vault until he learns a little more about the way things work.
Number 1 rule with bitcoin that nobody should have to learn the hard way is
ALWAYS store your coins
offlineLook into paper and hardware wallets OP.
If you need help checking that your Coinbase account is setup ok then ask their support for help, do not give your password to anyone!
You should start learning about different types of wallets for future use.
Set up a desktop wallet (Mycelium is easy to use) and send a small amount of BTC to yourself and back again. Remember that Coinbase will cove transaction fees but you will have to pay when using just about any other wallet, set the fee to about 0.0005 for fast confirmation as the mem pool has a backlog.