The general opinion seems to be that the lightning network offers good privacy for the sender but not the receiver. However, I will be both the sender and receiver. Would it be good practice to swap the coins on the lightning network with coins on the blockchain and therefore improve the privacy that way? How good of a solution is this compared to coinjoin or Monero?
The research is 2 years old now, but you can check
https://lightningprivacy.com for an overview of the weak spots in Lightning and the improvements that mitigate them. A powerful way to layer these tools is to coinjoin first, then open a Lightning channel with your private UTXO once you want to spend your coins.