While the achievement says "Get your first coin" this is not completely correct. It forgot to say get your first GOLD coin.
During your first play on a level you will notice some silver coins, this is not what your looking for here. In order to unlock this achievement you will need to collect a Gold Coin.
Gold coins appear when your playing that level in "Challenge Mode". To unlock the challenge mode for a certain level you will need to first beat that level and get an "A" rating. Once you get an "A" rating on a level simply replay it in the challenge mode and collect a Gold Coin to unlock this achievement.
To get an "A" rating you will have to do well in combat, collecting silver coins, and most importantly completing the level quickly (time).
The easiest level to do this in and the level I got it in is level 1.