Is it safe to store my coins (BTC & ETH) on the Coinbase platform or is it safer to download a wallet to keep them in?
I would recommend diversifying your btc holdings as much as possible. What if something happens to your wallet? That wouldn't necessary be better than coinbase. You could keep some in a personal wallet, some on several exchanges. Ideally by diversifying your capital, you can some what mitigate commonality of the risk. This prevents the possibility of a single devastating loss.
That said consider some alternative, if there is a single fault, but the probability is very low, that may be a good choice.
Consider https, which typically relies on one main root of trust who's security has been well verified, compared to some of googles authyoath crap (which places trust on a diverse sequence of relatively completely insecure mechanism) which have been hacked whilst https remains secure.
P.s. Google, stick to search engines.
Coinbase allows the purchase, sale and holding of three cryptocurrencies as of 7/4/17. Bitcoin (BTC), Ethereum (ETH) and Litecoin (LTC) are supported on the Coinbase platform and there is also a U.S. Dollar wallet (USD) where you can store U.S. Dollars from sales of coins or from transfers. Coinbase has insurance that protects every customer against the loss of digital currencies. If digital currencies are stolen, Coinbase's insurance company will make you whole. Also, if you are a U.S. Citizen, your U.S. Dollar wallet (USD) is protected up to $250,000.00 by F.D.I.C insurance. The USD is where you park your cash before a purchase or after a sale. I store 99% of my digital currencies on Coinbase and I sleep quite well.
I wouldn't do it.
In case you do that you trust all your Bitcoins to someone who may be hacked or go out of business.
I would keep my coins in cold storage.