If anonymity is your primary concern, exchanges that do not require KYC are the best, like Binance, Kucoin and Uniswap. However, verified accounts can enjoy the benefits of crypto cards that are issued by certain exchanges like Binance, Coinbase and Crypto.com.
All exchanges accept deposits of the cryptocurrencies they support. For FIAT currencies, most exchanges accept bank transfers, and some even credit & debit cards. Fees vary per country, method and currency.
As with deposits, all exchanges allow crypto withdrawals, albeit with a small fee. FIAT withdrawals are another story. Some exchanges may even not have such methods. Nevertheless, the primary method that most offer is bank transfer. Again, fees depend on the country, bank network and currency.
In terms of trading volume, the big players at the time of writing (July 2021) are Binance, Coinbase, FTX, BitMart, Kraken, Hotbit, Bitfinex, Crypto.com and Gemini.
For gambling transaction purposes, yes, Binance is a better option if you are not a US resident. Binance gives you the option to not proceed to a KYC process, and Coinbase generally speaking is way stricter with regulations.