Watch the Day One video for some ideas on how to start playing.
In-game, click the crown and try some quests. On the policy screen, click "Give Me Advice".
The Cheat Sheet has a great summary of the game without too many details.
Read sections one and two of the manual for more specific information.
You can visit the n00b Board in the forum. Check out this post on starting out, and this one on trading.
Brennon Adams (aka Iuewen) has posted a Let's Play: Let's Play Battle Mines - Episode 1 - The Basics
Try talking to your neighbours in the game, maybe they can help!