The blockchain is the shared journal of all transactions on the network. Mining is the process which authors new blocks to extend the blockchain. Miners do not need unconfirmed transactions to create a block, but they earn transaction fees from including them. The only transaction in a block that is required by the protocol is the coinbase transaction. (We refer to blocks that only have a coinbase transaction as empty blocks.) As the difficulty regulates the interval, mining acts as a distributed lottery to elect a block author about every ten minutes. As a reward for the expended work, the lucky miner collects the transaction fees of the included transactions and creates some new bitcoins for themselves. The long interval between blocks gives other network participants time to catch up to the new chaintip and converge on a shared truth.
First, I browsed the wiki to figure out how SN created the genesis
block to earn bitcoin. It said he published the Bitcoin Core software,
used it to create genesis block and was rewarded with 50 bitcoins.
The order of these events is mixed up. The Genesis Block was created before the Bitcoin software was published. Each block points at the previous block. Block 3 points at Block 2. Block 2 points at Block 1. Block 1 points at the Genesis Block. This forms the blockchain.
GB ⇐ 1 ⇐ 2 ⇐ … ⇐ 719935
The Genesis Block is special because there was no previous block. The Genesis Block was created to define the starting point of the network. The first release of Bitcoin included the Genesis Block to make all nodes agree on the starting point!
However, the Genesis Block includes the headline of a newspaper from that day. This proves that the Genesis Block couldn't have been created before the newspaper got published, which was only a few days before the software was released. Satoshi Nakamoto did not create any money for himself with the Genesis Block, because the output of the Genesis Block cannot be spent. Since the Genesis Block is hard-coded as the starting point (for the first block to have a previous block), every new node already knows about the Genesis Block, and they do not read the Genesis Block like other blocks. Because nodes do not read the Genesis Block normally, they don't notice the money the Genesis Block created. The first block that is new to Bitcoin nodes is Block 1. The first spendable money was created with Block 1, because the first transaction that nodes process is the transaction in Block 1.
Even empty blocks provide proof-of-work and secure the network.
Where can I see that public ledger?
You can acquire your own copy of the blockchain by running the Bitcoin software, or browse one of many blockchain explorer websites such as e.g. mempool.space, blockstream.info, or blockchair.com.