I'd chose Blockchain.info between those two. Coinbase allows you to send offf-chain transactions but they don't give you the access to your funds you need to trust them, do a google search of MTGOX, CRYPTSY and there are more cases where exchanges failed to secure the coins. Bitcoin is made to be decentralized, not centralized like coinbase does, you need even to provide your information to coinbase but in general you don't need to give out any information to anyone to be able to use bitcoin and other cryptocurrencies.

Blockchain.info on the other hand keep everything encrypted so that only you have the access with your password and ID to your funds. Please backup your seed as they give you an option for that, with that seed you are able to recover your funds if for some reason blockchain.info stop to provide their service.

Understand that you don't have to register anywhere to be able to use bitcoin. I'd suggest that you go and search on youtube how does bitcoin work, there are really great and well informative videos about this topic. Best would be in case you can to only keep the private keys within your device for extra security, once you understand what is a cold storage and what are paper wallets you might move there in case that you just want to hold your coins for a long term.