It depends if you want to buy on CEX or DEX...

If you want to buy on CEX (centralised exchange), the fees applied are the CEX fees (from 0.1% to around 2%).

If you want to buy on DEX (decentralised exchange), the amount of fee depends on with ewhich altcoin you buy, because it apply the chain fee. For exemple, if you want to buy BTC with ETH, you need to use the ERC chain (this is the most expensive IMO), if you want to swap MATIC to BTC, you need to use the Polygon chain, and so on... To use DEX platforms (like Uniswap, PancakeSwap, etc...) you need to use a non custodial wallet like Metamask, TrustWallet, etc

The cheapest chain is BEP20 (Binance Smart Chain), and the Layer 2 chains of Ethereum (Arbitrum, Optimism, zKSynk, etc).

About the speed of the transactions... on CEX the transactions are almost instant (if you use the market price), and on DEX you need to wait some minutes usually (my experrience is that around 90-120 seconds is the average time to execute a transaction).

I hope I could help you a little bit with my reply.

Have a nice day!