What size timber beams do we need to span our ceiling/floor system?
This question is a bit on (or over) the edge of what's sensible to ask here. Even those who might be structural engineers (I'm not) can't do your structural engineering over the internet. People get hurt when buildings fall down.

Hopefully you have some big trees available.

Running it through a basic wood beam calculator (DAGS, which will keep any links from getting old), the 20 ft span on 30" spacing will need to be at least 4x16, and if you are sawing (or hewing) it yourself you're going to have to learn to grade your lumber for beams (a knot or split in the wrong place can make a much larger beam unsuitable for the load, depending where it is on the beam.) Or you do what the old-time old builders of your old oak log home would have more likely have done, and put two posts in the room supporting a central beam, rather than trying to span 20 feet. Few barns even had spans over 12 feet...

If you mean "tulip poplar" AKA tuliptree (wood tends to be greenish), it might be usable. It's a somewhat respectable wood. White poplar (young bark can be greenish, but the wood is dead white) aka cottonwood aka popple -  I believe is generally considered unsuitable for beams - it's a lovely soft white wood to carve, you can cut wall panels from it, but it's weak as can be. Pine or oak are both fine.