Earlier quoted context omitted.
> As a general rule, don't use conventional encryption methods to transmit information with long-term sensitivity (e.g. medical records, etc.). So what do you suggest I use today , given that I have a medical record here and I need it over there?
Most recent NSA transitional recommendations are for 3072-bit asymmetric, and AES-256 symmetric. Details: https://www.nsa.gov/ia/programs/suiteb_cryptography/index.sh... For network transfers, you'll likely also want to select your encryption with PFS: https://en.wikipedia.org/wiki/Forward_secrecy
PFS will make it so connections need to be individually attacked but since most PFS is done with Diffie-Hellman variants (susceptible to Shor's algorithm), the group size also needs to be large enough to resist early quantum computers.
For defense against large quantum computers, different PFS schemes need to be used (fortunately not hard to construct from other post quantum primitives).