At thr end of the day you can't be anonymous with bitcoin. You can send and receive from your own wallet without having to give personal information, and have full control over your crypto, but anywhere it touches other monetary systems (like coinbase), you can be tracked. If that makes sense.
I'd say Exodus or Coinomi for a basic software wallet that can hold both bitcoin and doge. Just make sure to backup the keys/phrase to generate the wallet again. Write it down somewhere so even if your computer/phone dies, you'll be able to get access again to the crypto.
Lightning nodes will obfuscate your transactions. Combined with a wallet that ALWAYS uses new addresses... And it would take. National resources to sort it out.
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