How do I know which exchanges are serious actors? How do I find bitcoin exchanges? 
How do I know if the exchanges can be trusted?
For finding Bitcoin exchanges, check here (exchanges are ordered by volume):
https://www.cryptocompare.com/coins/btc/markets/USD

There are two types of Bitcoin exchanges to consider:


Centralized
Decentralized


With decentralized exchanges, users keep their assets local, and so there is no need to trust a central authority. At the present time, it is harder for decentralized exchanges to build up liquidity. This means that the trading volume is lower and the bid/ask spread is wider.

Here are couple of resources regarding decentralized exchanges:


0x Protocol for building decentralized exchanges for Ethereum
assets
Waves platform DEX - decentralized exchange
https://www.cryptocompare.com/exchanges/guides/what-is-a-decentralized-exchange/


Now, with centralized exchanges, the central authority running the exchange must be trusted. The process of deciding who to trust is less straight forward. Here are some attributes to consider when evaluating a centralized exchange:


The people running the exchange or company
The country where the exchange is subject to regulations
The security practice the exchange uses to protect customer assets
The insurance policy offered by the exchange
The history of customer issues at the exchange, reviews etc.
The transparency the exchange offers
The communication facilitated by the exchange with its customers


I have put together this YouTube playlist discussing some of these attributes with respect to GDAX, Coinbase's exchange: GDAX and Coinbase cryptocurrency exchange - Overview