If Coinbase is not working for you, check out Kraken[1] and Gemini[2]. As far as I know these three are currently the most reputable in most countries they support.
Also, second question, If I have a Ledger Nano S, can I buy coins online and store them on it ?
Yes. Send over the funds from the exchange to your Ledger Nano S' bitcoin wallet address. To prevent confusion on your side, please read this post concerning wallets:
https://bitcointalk.org/index.php?topic=1631151.0Or do I have to go to a local Bitcoin ATM ?
You go to a bitcoin ATM to buy or sell bitcoins, not store them. With bitcoin, you are your own bank.
[1]
https://kraken.com/[2]
https://gemini.com/