How am I supposed to know if I am in profit or a loss if I have been trading across multiple exchanges?
You should build and track your trading portfolio there are some tools that you can use to track them.
Sample Binance you can follow the
guide on exporting your all trading history from Binance. Once you have done on exporting your trading history from binance you will need a tracking tool to calculate your all profit and loss including fees like this tool
https://cointracking.info/import/binance/index.phpFor example, if I bought $1,000 in Bitcoin on Coinbase, transferred it over to multiple exchanges, and purchased other coins with that $1,000. Is it as simple as downloading the tax reports from the exchanges?
It's simple but it is not simple as downloading tax reports. If you follow the guide carefully from the link above you can calculate all of your profit, fees, and losses using the tracking tool above.
You can see all supported exchanges from their front page here
https://cointracking.info/ they also have old announcement thread from here
https://bitcointalk.org/index.php?topic=220238.0