What is the best exchange and how do I make a buy order there?
we need more information.
what country are you located in? are you looking to deposit fiat money, or are you just looking for a place to trade? exchanges are very jurisdiction-specific and many do not support fiat/banking, although they may support stablecoins.
if you're in the USA: binance.us is probably the cheapest option commission-wise (0.1%) among fiat exchanges. for non-fiat exchanges, kucoin is probably your best bet at 0.1% since binance.com doesn't allow USA customers.
if you're in europe: kraken has the most liquid BTC-EUR market and is very competitive fee-wise. for altcoin/stablecoin trading i would recommend binance.com since they have the best liquidity and 0.1% trading fees. there are slightly cheaper exchanges (like poloniex, whose fees start at 0.09%) but binance has better liquidity and reputation.