I've been allocating some savings to BTC. I keep seeing exchanges fall, and not your keys not your coin. I've been using the Coinbase wallet app. Is that safe?
If it is not a cold wallet which you own the wallet's private keys, then it will never be "safe". Coinbase wallet app is an "app". It is connected to the internet. App's can get hacked. Operating system's can get hacked. Plug-in's can get hacked. The only way to stay safe is creating your wallet when you are offline, preferably on a fresh formatted linux OS computer which don't have a wi-fi adapter on it and then writing your keys down on a piece of paper.
If Coinbase goes down am I out?
That's how it works with the exchanges. They go down, you go down.
I need to look into a cold wallet but for right now am I good?
Nobody knows what kind of crap you'll wake up to tomorrow.