How can I know if a website is trustworthy ?
First, here's a list of bitcoin exchanges:
https://bitcoin.org/en/exchangesYou don't mention what country you're in and this detail may impact which exchange you can/want to use.
As for whether it's reputable, one way to find out is once you've selected an exchange search for posts on this forum related to that exchange. If you get several, generally positive mentions about the exchange you
could assume it's reputable. If people have had problems you'll probably see threads about it here. And if there's no mention of it on this forum at all, proceed with caution!
I use Coinbase for BTC, ETH, and LTC. Coinbase operates in many countries around the world and they do charge fees for their services but they are mighty convenient. If you're willing to wait a few days for the transaction to settle (meaning having access to your coins to spend or transfer to another wallet) you can do a purchase direct from a dedicated savings or checking account and the fees will below, less than 2% (whereas using a debit/credit card will be 3.99%).
Good luck to you!