I find the whenever I start the game, the first handful of explorations determine whether I am going to be able to enjoy the rest of the content or not, so it's entirely luck based, or is there a way I can beat the odds?

It's very much luck based, the game blends Survival/Resource Management and Rogue-like elements. However, as you go along you get more items which give you greater control over your resources, like the Fishing Rod to get food whenever you want.

Explore when you can, try to find the hidden characters as soon as possible. 

Pro tip; it only takes one board to block an invasion. So instead of slapping all your boards up at once (which takes time and boards) you can just use one, and replace it each time it falls.