Is it safe to store all my coins on Coinbase? Is it safe to store my coins (BTC &amp; 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.