Hardware wallet can be used anywhere? I haven't yet traded any cryptocurrencies. I'm planning to buy BTC through Coinbase then exchange BTC for a small amount of XRP.

The main reasons for going with Coinbase is it's safe, well regulated and I'll always want an account there. For XRP, I'll use changelly.com since they are popular, have low fees, and haven't been hacked (yet).

A big concern is safety. Coinbase probably has a good wallet but I'd like to use a hardware wallet since I'll be using other exchanges for various alt coins not offered on Coinbase. I don't really trust other exchanges and hence the hardware wallet.

Is a hardware wallet the safest option in this case? Are there any issues using it at Coinbase and Changelly?

Do I still need to setup wallets at exchanges I'm using?

When using a hardware wallet, from what I understand, you plug it into your computer via USB and do whatever transaction at whatever exchange. Once complete, you unplug it. From this, it basically can't ever get hacked?
Hardware wallets are easy-to-use ways (compared to paper wallets (hard to use) and software wallets (not as safe as hardware wallets)) of being in control of your private keys hence the safest and the easiest way of using and storing cryptocurrencies.

When you use a (traditional) exchange as a wallet, you won't be in control of your private key so the exchange can steal your funds or get hacked and lose them etc.

There are decentralized exchanges where you don't need to deposit your funds (thus you don't need to trust, a.k.a trustless) like RadarRelay. You can use them with software wallets and some of them with hardware wallets directly. 

Don't forget that cryptocurrencies' reason of existence is being trustless, don't hand over your funds to anyone, always be in control of your private keys.