Step 1. Purchase Bitcoin:
As of 1/11/2018, the best place to purchase Bitcoin (BTC), Bitcoin Cash (BCH), Ethereum (ETH), or Litecoin (LTC) directly with cash or some credit cards is at https://coinbase.com/
Notes: Pay attention to any notices of potential delays in transactions do to high trade volumes on any of the currencies you are considering purchasing. Also, be aware that Bitcoin currently has very high network transaction fees until they address the scalability issue. Expect to pay the equivalent of around $20 in Bitcoin transaction fees to send Bitcoin to an external wallet.
Also, If you are looking to purchase an alt-coin like DASH, for example, I recommend buying Litecoin at Coinbase and sending that to an exchange where you can either directly trade it for another currency, or you can trade it for Bitcoin before trading that Bitcoin for something else.
Except for those listed above, most crytocurrencies cannot be purchased directly and must be traded for Bitcoin instead. However, some exchanges do allow direct trading with some alt-coins such as Litecoin or Ethereum instead of Bitcoin.
Step 2. Send Bitcoin to your personal wallet to store them, or send them to an exchange for trading:
While Coinbase does privide a wallet for storing your Bitcoin, it isn't recommended to store your Bitcoin, or alt-coins, on any exchange or online wallet where you do not have complete control and sole ownership of your private keys.
For a list of Bitcoin wallets, along with info regarding their benefits and level of security, go
I recommend installing Coinomi if you have Android, or Jaxx if you are an iOS user. Those two wallets allow you to store many different cryptocurrencies, not just Bitcoin.
Step 3. (Optional) Trading Bitcoin for Alt-coins:
If your goal is to trade your Bitcoin, or one of the other alt-coins you purchase at Coinbase, for a different Alt-coin, you'll need to create an account with an exchange like Poloniex. First, send the Bitcoin or alt-coins you purchased from Coinbase to the appropriate deposit address that Poloniex gives you. Then, you may either trade Bitcoin directly for another coin, or you can trade the alt-coins you purchased from Coinbase for Bitcoin before trading Bitcoin for a a different alt-coin.
I know trading an alt-coin for Bitcoin so that you can trade Bitcoin for another alt-coin is inconvenient, but that's just how it works at the moment. In truth, however, you can trade some alt-coins directly for a limited number of other alt-coins at some exchanges. At Poloniex, for example, you can trade Ethereum directly for some of the other alt-coins. So, if you wanted, you could buy Ethereum directly from Coinbase with fiat money, send that Ethereum to Poloniex, and then trade the Ethereum for another coin.
Notes: Most big exchanges in the U.S. require some form of identity verification, and some exchanges may take a few days to verify your identity. Other exchanges, however, can verify your identity right away by asking some personal questions only you would have the correct answers to.
If you would like addition info such as market trends, wallet links and coin websites, as well as which exchanges allow you to trade particular alt-coins, most of that information can be found at https://coinmarketcap.com/. Addition details, such as coin specifications, can also be found at https://www.cryptocompare.com/.