Slitherine

I can pick any campaign battle and play it but how do you start a campaign. I want to play the battles in order and carry veteran troops over and gain new tech. Is this possible with the beta?

When you enter new game from main menu, scenario tab is selected. Switch to Campaign tab to have options related to campaigns.

I was playing from the campaign menu. When the battle was over I was back at the main menu with no prompting for the next mission or re-fitting my army.

What campaign were you playing? Tutorial, 1939 or 1941?

1939

How Poland scenario went trough? Did you win decisively, by marginal victory or did you lose?

I went trough cheat codes to test it quickly and it works fine for me. After lose it gets you back to main menu because game is over. Marginal leads to Low Countries and Decisive to Norway.

I thought I won with one turn to spare. But based on your description, I must have lost.
This brings up the fact that I have never seen any indication of victory type.
Furthermore, the victory conditions are not shown. I assume I need to cap the gold bordered cities?

Victory conditions are not mentioned but will be in the briefings (there should be test placeholders for briefings).
To win you need to capture objectives with gold border that is right.
After finishing scenario there should be a screen with picture and some text indicating result and option to proceede or view battlefield.