By mining, you can earn cryptocurrency without having to put down money for it. That said, you certainly don't have to be a miner to own crypto.  You can also buy crypto using fiat currency (USD, EUR, JPY, etc); you can trade it on an exchange like Bitstamp using other crypto (example: Using Ethereum or NEO to buy Bitcoin); you even can earn it by playing video games or by publishing blogposts on platforms that pay its users in crypto. An example of the latter is Steemit, which is kind of like Medium except that users can reward bloggers by paying them in a proprietary cryptocurrency called Steem.  Steem can then be traded elsewhere for Bitcoin. 
In Person: Over-the-counter platforms such as CoinCola or LocalBitcoins are resources to find people in your area to trade bitcoins with. Trust and security can be a concern, which is why it's recommended you transact in a public place, and not necessarily with large amounts of cash. Some of those platforms, such as CoinCola, will allow its users to upload an ID proof. In this case, you will be able require the ID proof of your trade partner for added security.

Lawbreakers have to hide and camouflage the money gained from their exploits. Currently this is done with fake bank accounts, gambling, and offshore companies, among other stratagems. There are a lot of concerns regarding the transparency of cryptocurrency transactions. But, all of the necessary regulatory elements, such as identifying parties and information, records of transactions and even enforcement can exist in the cryptocurrency system.
Though transaction fees are optional, miners can choose which transactions to process and prioritize those that pay higher fees.[69] Miners may choose transactions based on the fee paid relative to their storage size, not the absolute amount of money paid as a fee. These fees are generally measured in satoshis per byte (sat/b). The size of transactions is dependent on the number of inputs used to create the transaction, and the number of outputs.[3]:ch. 8
Now, if there is no central system, how would everyone in the system get to know that a certain transaction has happened? The network follows the gossip protocol. Think of how gossip spreads. Suppose Alice sent 3 ETH to Bob. The nodes nearest to her will get to know of this, and then they will tell the nodes closest to them, and then they will tell their neighbors, and this will keep on spreading out until everyone knows. Nodes are basically your nosy, annoying relatives.

On 3 January 2009, the bitcoin network was created when Nakamoto mined the first block of the chain, known as the genesis block.[19][20] Embedded in the coinbase of this block was the following text: "The Times 03/Jan/2009 Chancellor on brink of second bailout for banks."[10] This note has been interpreted as both a timestamp and a comment on the instability caused by fractional-reserve banking.[21]:18