the Green address app for my phone. Would this be a safe way to go
You have a point though since blockchain wallet have been used by most users here on bitcoin world but still you should put only small amounts because risk of lossing is still there. If you have lots of bitcoin its better to use desktop wallets which is somehow safer than web wallets. and also web wallets is more faster when it comes to online transaction and other web wallets out there.
I've been using blockchains wallet since I've started and I'd recommend it to anyone that's not planning on keeping large sums of btc on their wallet, if you're planing on so, I'd suggest you to get some offline wallet and more secure one.
Correct, blockchain.info is somehow a famous web wallet as OP mentioned on green address i do heard of it also but i would not intend to use it ,i prefer the most reputable web wallets though. Offline wallets are better for huge amounts to increase the security of your bitcoins.
Next time we should be very careful when we are logging in into web wallets sites because there are lots of clone sites which you could hardly notice it when you dont pay attention to it and for sure you would be a victim on that site because all of those are phishing sites which they actually get your logging in details and could able to stole your bitcoins on the actual wallet.