Fire Emblem: Blazing Sword, also known as Fire Emblem: Rekka no Ken, though titled as "Fire Emblem" outside of Japan, is a good starting point. Among the easiest games of the series, alongside being, quite simply, a fun game. It also has no major gimmicks, thus it will be quite easy to transition between that and other Fire Emblem games.
Fire Emblem: The Sacred Stones is also a very valid choice. It is quite easy, easier than Fire Emblem: Blazing Sword, though still remarkably fun. To make it more forgiving for a newcomer, infinite grinding is an option, and can help you become more accustomed to properly leveling a team.
Fire Emblem: Path of Radiance is another fantastic starting point. Again, it is among the easier games within the franchise, and you can learn the story of Ike, the famous Radiant Hero, and a playable character in the Super Smash Bros. franchise. It features a "Bonus EXP" system, which the player is given for preforming well. The player can then award this EXP to certain characters in order to help them level up. This makes the game quite easy, and, similarly to Fire Emblem: The Sacred Stones, can help you learn how to properly train units and build your teams. The issue, however, is that this game is incredibly expensive, often reaching a price of sixty United States dollars (or €51.84) or so. It can, however, be easily emulated.
Fire Emblem: Shadow Dragon is also a very good starting place. It also has no major gimmicks, similar to Fire Emblem: Blazing Sword, though it is perhaps simpler and features Marth, the Legendary Hero-King, who you may be familiar with as that character in Super Smash Bros. You may enjoy learning about Marth's back story and origins, which can be quite easy to do, as Fire Emblem: Shadow Dragon is currently among the more common Fire Emblem titles in North America (and Europe, I believe), and is a remake of the very first title in the series.
As for Fire Emblem Awakening, only play it after you have played two Fire Emblem games or so. It is radically different from other Fire Emblem games, to a point that it will most likely give you certain expectations for the rest of the series, and will most definitely lead to disappointment when every game is so incredibly different. It will also lead to several poor habits which could effectively make you worse at the rest of the games in the series. It was my starting point, and some of those bad habits still haunt me.
TL;DR: Fire Emblem: Blazing Sword (the Fire Emblem which you mentioned), Fire Emblem: The Sacred Stones, Fire Emblem: Path of Radiance, and Fire Emblem: Shadow Dragon are all fantastic starting points. Of those, I personally recommend Fire Emblem: Blazing Sword the most.